Even in the summer and when there are boatloads of tourists, Prague can accommodate them so you don't feel like you're in lines all the time and there's a lot of energy with the wide-eyed fun positive, often gullible and noisy, tourists.  But if you don't mind sharing a big bench table in some of the few pubs that still have them - you could easily be with people from 5 different countries. 

Find a balance between getting around and how many things you can see.  You don't want to spend all your time getting to places.  If you can extend your time, Vienna is 4 hours away by car , 5 by train - and it's a nice ride.  It's a magnificent city with two incredible royal palaces (go to Schonnebrun - it's every bit Versailles and actually looks modelled after it), Dresden is even closer and very interesting as it's rebuilding itself.  The awe of what happened there is striking, but it's now very beautiful.  (A day return from Czech railways eshop costs 22 euros per person).   Other destinations, Berlin, 5 hours by train. Budapest, 8-9 hours, and Krakow 10 by train (or 8 hours train/bus  through Cieszyn/Český Těšín) or Cieszyn/Czech Cieszyn 4 hours by train, start to get longer, but you can fly anywhere very fast, Paris, London, Amsterdam, etc. June might be a little pricey.  The trips in Czech down below are quite enough.   Easyjet and SmartWings landing Prague 

Taxis are now pretty consistent, but still best to call one when possible (AAA phone #:  dial 14014). 
Czechs speak English to a remarkable degree and learned it rapidly considering it was compulsory  to learn Russian up until '89/'90.  Czech language is rated the third most difficult language in the world which makes learning anything else (besides Chinese and Estonian) easy for them.   (lots of people like to fight about – OK, 5 th )  4-5 languages is common . 
Here's a sample – most relevant:  Beer is pivo (pee-voh - ironically enough),
-one beer, jedno pivo (yed-noh),
-two, three, four beers, dve (duh-vee-yea) piva,
-five beers pet (pee-yet) piv and after
Beer is average a half liter - that's a big glass - for 25-35Kc (Crowns or Koruna), that's about 1 - 1.5 euro.  The pubs are cheap, too, for food.  A real pub shouldn't cost more than 200Kc each for two beers and meal - about 8 euro.  

Knowing a few polite words is GREATLY appreciated, 'though.    Prosim (proh-seem) is 'please' as well as 'you're welcome', Dekuji (day-koo-ye) is a polite 'thank you', jidelni listek (yih-dell-nee lees-tek) is menu, Pardon and sorry work, as does toilet, OK, taxi, policie and strangely blah, blah, blah.  Kde je is 'where is' (guh-day yeh), pozor is caution, hrad is castle... 
Czech food is made to hold down the fort for voluminous beer consumption and Czechs are fiercely proud of their food (which I find incredible - you'll see).  Gulas(h) is meat in gravy with bread dumplings and can be great (once a month), Rizek is like a schnitzel - pounded flat meat -maso- (hovezi is beef, kureci is chicken and veprove is pork) breaded and fried, great with lemon and mashed (bramborova kase) - it's worth having.  Certainly is great with beer.  

Big loud beer mug clinking pub owned by Pilsner Urquell is U Vejvodu - excellent quality Czech fare with some creativity – near the center between Old Town and Narodni Trida, Pilsner has a few pubs around, Kolkovna is also good just off Old Town.  Pozor, though, Pilsner is a 12 degree beer (dvanactka - , and while great with food, for the less than seasoned beer drinker, it can get you pretty jammed if you have a bunch of them.  Other pubs almost always have a 10 degree beer as well, which is plenty good for the long-haul.
Anywhere decent enough and certainly anything in the center, tips of 10% are welcome, but don’t reward any “service with a sneer”.
Lots of pictures to be taken.  It is magnificent.  You can walk the basic highlights in a full day and most guidebooks have prepared treks varying by distance/time/interests - some only highlight pubs with castles and churches used as landmarks to get you there.  It's best to walk as cars/buses can't get into the nooks, crannies, tunnels, passages, etc. 
Wenceslas Square - Vaclavske namesti, huge, beautiful, where all the anti-commie demonstrations were, Havel's speeches, where the Russian tanks rolled in in '69, Muzeum is a beautiful building, but it's mostly stones and bones, so much art was stolen from here by Nazis , shopping here, pretty cafes, and at the bottom take a right on Na Prikope (crystal and glass are of the highest quality and such a good value - these Bloomingdales for 3.5 times the money, also garnet and gold are very good here, marionettes, creepy European style, wood toys) en route to->
Namesti Republiky - Obecni Dum - Municipal House - stunning classical Art Nouveau, have lunch downstairs in the lushest "pub" in Prague - Czech fare & reasonable, or have a coffee in the unreal cafe upstairs - not so reasonable, but worth it . The concerts are very good and Smetana Hall almost eclipses the music (Opera at the Estates Theater is also a screaming deal (pun intended) pref Mozart as he conducted the premier of Don Giovani there and was often a resident, as was Beethoven - on old square, Maltezke nam (Maltese Sq - what a sexy address!)->
Old Town, Staromestske namesti - go into Tyn church, dramatic gothic with two buildings stuck right onto the front of it, behind it is a beautiful court yard, Ungelt, with shops and cafes  - directly through the courtyard is Chateau Rouge (only the name in French, elbow to elbow NY style bar, always turns out to be fun, bit down and dirty), back to the square, St. Nicholas cathedral, baroque, lots of pubs around , Parizska (Paris St) is beautiful and posh for cafes and jewelry, art.  Pravda restaurant is pricey but fun in that each menu selection represents a meal from a different country - great to share, the astrological clock is here, the king regarded it as so beautiful that he had it's maker's eyes cut out so he couldn't make anything more beautiful, interesting, but remember it's just a clock when you go to see it do it's little thing - people always seem to think it's supposed to do more, jazz club Zelezna right there on the street of the same name ->
While in Old Town, don't forget to visit Jewish Prague, including synagogues, cemetery, and museum. The area is called the Jewish Quarter or Josefov.
There's a windy tourist walk that sheer tourist volume will suck you into (directly left if you're looking at the clock) with many good shops, garnet and gold, puppets, art, glass/crystal and every kitchy souvenir thing you could imagine (nesting dolls are NOT indigenous), Reykjavik might have the best seafood in Prague  ->
Charles Bridge (Karluv Most) The goods being sold on the bridge are really super and a good deal.  The photography and paintings are very reasonable and it's a nice place to remember where you bought it.  You are now under the castle and the walk up really takes you into another world.  The tower on the Old Town side is open and it's worth a climb to see the whole city laid out, but you will also be afforded many views from the castle, too.  There is a staircase on the left before the tower that goes down to Kampa island, very nice and the park is often full of hand-drummers and dancers and has that smell in the air.  Take a left on the Mala Strana side after the tower and go to Maltezke namesti, beautiful square loop to your right and the big yellow building behind the looming John the Baptist statue is Na Satre Posta - the first post office in Czech, down the street/alley to the right, opposite is U Maleho Glena (At Little Glen's) which is usually a great place to meet some like-minded folks and the music downstairs is very intimate and often excellent  Malostranske namesti is a beautiful square, castle everything, any guide book will tell you what's up and it will all unfold for you.  Going toward the castle on Nerudova on the right is Cowboys, very nice garden after a wild twisting way through the restaurant.  Treat yourselves to a dinner at Palffy Palace - Not really even that pricey.  From there, it's ALL castle.  The tours can be good and better than the headsets.  Do try to read up on the history of Czech - it's a reflection of all Europe.  This place was the center of wealth in Europe twice.  It was the Czech and Moravian border regions that Hitler annexed following the Munich Agreement that was the beginning of WWII.  It really was a land of kings, too.
Roxy and Radost are the two main electronic houses, but you really can't avoid it these days.  There are tons of event listing free magazines all over for specific things with details about what's up.  Keep your eyes open for concerts now, so you could get tickets in advance.   Lots of jazz, Rock cafe is only OK, Vagon occasionally putting on something great .  There are web sites for that, too.  
  Internet is available all over - lots of wi-fi. 
Trains are fun.  It depends on your itinerary and having a car is nice, too.  Driving is not bad, rolling hills, easy directions, but that stupid two lane passing nonsense is a bit hair-raising. 
Karlstejn is a very convenient day trip from Prague, less than an hour by train from Hlavni Nadrazi (Main Station) or Smichov, both on the metro (public transportation is amazing here, trams, buses and underground, and super cheap.  Round trip is about $5.  Very nice trek up the hill, lots of good pubs for chow, particularly this game restaurant behind the castle - as you exit the castle, take the path in the woods to the left and when you come down follow the street in the same direction and it's on the right - can't miss it, it's the only thing there. 

Go back down the street and it will bring you back to the town.
You can do a little reading, but a magical romantic place you MUST see is Cesky Krumlov (by bus through Ceske Budejovice - where the original Budweiser comes from, or by car or by direct Student Agency Bus from Na Knizeci).  Do the castle tour, just soak it in.  Very nice hotels.  You can canoe the river, too.  Get the wider rafts or you'll fall in a lot.    The best spa town is Karlovy Vary, which also hosts a very famous film festival, and more information about it is below.  Also feels like the domain of kings.   Finally, if you want to see something beautiful and bizarre at the same time, Kutna Hora (car/train) is where all the silver was mined.  The mine tour is a real eye-opener when you see what they dealt with and you'll experience dark like nothing else. 

Well worth a visit on the way to Kutna Hora is the Ossuary, or `Bone Church` at Sedlec which is elaborately decorated with human bones from an overflowing cemetary, made too popular when it was sprinkled with soil from the Holy Land - macabre and fascinating. 

For pictures and further details : Ossuary at Sedlec. All three are Unesco Heritage Sites.

Details on the spa town of Karlovy Vary:

Karlovy Vary is easily accessible via bus (not the train, for some reason it takes twice as long). Buses leave from Florenc Terminal (near Hlavni Nadrazi Train / Metro Station) roughly every hour, and also call in at Dejvicka Bus Station and the airport on the way. The journey time is 2 hour 15 minutes, but unless you get caught in traffic you can knock off between 15-25 minutes.

The cost is reasonable (155Kc each way at time of writing) so an excellent alternative for the independent / budget traveller who doesn't want to go on a formal tour.

Once there, getting around this famous spa town is easily done on foot. Just head down the main street from Karlovy Vary Terminal to the canal region (in fact it is a river called Tepla). The best bet is to just follow the path of the canal which takes you past areas of interest such as the Thermal Spring, the Mill Colonnade, the Regional Museum, the Grand Hotel Pupp and a heap of boutiques and gift shops.

Once you have reached the Grand Hotel Pupp, catch the cable car to the Diana tower and enjoy a sensational 360 degree view of the entire city and surrounding area. The tower features both stairs and a lift for the mobility impaired. Nearby the Diana Restaurant offers a place to have something to eat and drink, or alternatively go for a coffee and cake in the Hotel Pupp Cafe.

Karlovy Vary is not only famous as a spa town, but also for Becherovka, a traditional herbal liquer made only here to a recipe only known to few individuals, so its worth a sample while your here.

Terezin is a town about 60Km from Prague. It is a fortress town that was used as a transit and prison camp during the Nazi occupation of Czechoslovakia. The prison is now a museum and a Jewish Museum stands on the town square. Both are a heartrending experience but the town's layout and other history, coupled with a short (50 minute ) journey by bus through the Czech countryside is a rewarding half or full day. The fare is 88 CZK (3.5 Euro ) each way by bus ( about 110 CZK by train ) and it's 200CZK for all the museums. That's far less than the tours charge. Litomerice is a larger town, popular with tourists, just 5 minutes away by bus, so a full day out combining both locations is worth it.  The bus departs from Nadrazi Holesovice, which is on Metro Line C, about 10 minutes from central Prague (Muzeum or Mustek/change at Florenc). It currently ( Dec 2011 ) departs from Stand 7 on the Metro Side ( opposite the station ) on the hour roughly.

 If you feel like visiting a brewery, check out the little town of Pilsen. It's a quick and easy bus ride (and cheap!) that is roughly an hour from the Zlicin bus station at the end of the B (yellow) line. I recommend the Student Agency bus - roughly $5USD each way and it's very comfortable, comes with free drinks, on-board TV/music entertainment, and an attendent that speaks both Czech and English. Going to studentagency.eu can give you a good idea of the timetables and everything. Once you get off at the station, there is no need to catch the tram into the main town unless you don't want to walk about 10 minutes. If you swing a right after exiting the station where all the bus stands are (facing the main road), then you just walk straight down the road and you'll arrive at the town centre. From there, there is a nice square in the center with a church, town hall, and a nice seating area with some fountains. The information center is also located right there and is very helpful. Grab a map and they will direct you to the Pilsner brewery (5-7 min walk) where you can get a nice 90 minute tour which includes a good .33L sample at the end. Very fun!