You can buy the tickets from machines or from the "Midori no madoguchi" (Green Window) at any JR station or travel agency.
You can save a little by buying at a discount ticket shop.
You can save even more by taking an overnight bus from Shinjuku Station. About 4300 yen each way. 7 hours.
Also consider a JR unlimited use pass. Buy it before you leave for Japan. About 30,000 for one week.
The price of the slower Hikari Shinkansen (Bullet Train) is 13,220 yen each way. This train takes 2hrs 45, while the more expensive Nozomi takes 2hrs 22. When I am in Japan I buy the 7 day Japan Rail Pass, assuming that I am spending some time in Tokyo where the pass is not needed. The 7 day pass costs 28,300 yen, only slightly more than the return fare to Kyoto and it can be used for other trips you may want to make to places such as Osaka and Nara. You need to buy the voucher for the pass before you leave the US and exchange it for the actual pass when you arrive in Japan.
Most good hotels can provide you with tickets and seat reservations but it is also fairly easy to do this at Tokyo station. Just follow the signs for the Shinkensen Tracks and it will not be difficult to find the ticket office. I have always found that staff have sufficient English to make to make buying the ticket or seat reservation trouble free.
I hesitate to say that anything in Japan is easy to do for a foreigner but getting from Tokyo to Kyoto is as easy as it gets. If you have not been to Japan before just allow plenty of time for the mistakes you will make and don't be afraid to ask for help. Almost all Japanese will go out of their way to help you.
How long in Kyoto? Well, Kyoto is basically as ugly as all other Japanese cities but it is stuffed full of shrines, temples, gardens and palaces, enough to require perhaps a year to visit all of them. Assuming that you won't want to visit more than four or five examples of each you would probably be suffering from temple fatigue after three or four days and be happy to move on to somewhere such as Osaka where most buildings are not much more than about 20 minutes old. In fact, many of the aforesaid temples are not much older but the Japanese have the ability to forget that a 500 year old temple has been burnt down and rebuilt 12 times.
I have been to Japan more than 40 times and am an expert on getting hopelessly lost and finding the way again, so get in touch if you need to know more. It is a really great country and, to my mind, the centre of world civilisation.
I agree with LondonBob re JR Rail Pass. The best place to buy JR Rail Pass in NYC is IACE Travel, in my opinion. They use very favorable exchange rates. You can just call them at (212)972-3200. They may answer the phone in Japanese, but can speak English.
BTW, I've bought my tickets from JFK-Narita at IACE Travel and HIS many times. They both are Japanese travel agencies, specialized in discounted airline tickets. They often offer better deals between the US-Japan rather than Expedia, Orbitz, Travelocity, etc. So, you may want to check out if you call IACE. HIS's phone number is (212) 599-4280.
I would recommend that you stay in Kyoto at least 2-3 nights without knowing who you are. Whether you should stay in Kyoto longer than a few days really depends on who you are and what you would like to see or do in Kyoto/Japan. For instance, younger people tend to enjoy shopping or partying in Tokyo/Osaka over Kyoto. If you are interested traditional and cultural Japan, Kyoto is much better than Tokyo/Osaka, and so on. While you are in Kyoto, you can take a day trip to Nara.
One place I would strongly recommend to first time visitors (besides Tokyo & Kyoto) is Hakone/Mt. Fuji. Hakone is part of the Fuji-Hakone-Izu National Park, less than 100 kilometers from Tokyo, and is very famous for nice "Onsen," hot spring spa. After spending time in Tokyo and on the way to Kyoto, you can stay in a nice ryokan with Onsen in Hakone for 1-2 nights. You may be able to enjoy seeing "Mt. Fuji" from there if the weather is nice. Please check out the following websites for further info.
I also recommend that you take a look at the old thread if you haven't.
If you are a non-smoker, you may want to book a non-smoking seat. Shinkansen's non-smoking seats are limited, and more popular than smoking seats. How to make a reservation in advance is in the thread I just mentioned.
I recently spent two days in Kyoto and I also did a one-day trip to Nara. Two days in Kyoto will let you realise what Kyoto stands for and its cultural and historical importance. As someone said you can spend weeks in Kyoto if you want to. We stayed at a budget Ryokan (hotel Lida) just in front of the Kyoto Station and it was very conveniently located. The Bus Station is also located at the main entrance to the Station and Kyoto sightseeing is generally done thro' the bus network. At the bus station you can get an English Map detailing all the routes and which bus no to take when visiting a particular temple or a shrine. When you buy the 500yen/day buspass you can use any number of trips/stops on that day. English announcements are made at every busstop and you can't make a mistake, Even if you do just hop on the bus again or ask someone for directions. The Japanese are so helpful you cannot believe until you experience it. If they can't tell you the directions properly they will literally take you there!.
I saw some tourists on bikes which I think perhaps is the best way to go around Kyoto. May be someone may have the experience to advise you.
Just to answer about the biking Kyoto idea.....when I am in Japan with high school students they always spend one day with rented bikes. They have so much fun! They say it gives them a chance to literally get lost, find their way, ask directions, and become completely immersed in Kyoto. The rental fee is very inexpensive and your hotel employees can tell you where to find the nearest rental shop. If you like biking, I recommend it!