You don't have to take transportation to see all the main sights in Salzburg - most everything is within walking distance, as central Salzburg isn't exactly a teeming metropolis compared to other European cities. And you definitely don't want to drive - you'd probably never find parking if you attempted such a feat. Plus, lots of central Salzburg is filled with pedestrian thoroughfares, so it'd be silly to even bother renting a car if you're only planning on staying in this locale.

However, if you'd rather hop on a bus or the train than hoof it, Salzburg's transportation is fairly efficient and inexpensive.  For one ride in the inner zone the cost is €1.70 and €2 for the regional zone. If you really want to stretch your pennies (or Euros), grab one of the Salzburg Cards - these discount cards allow you to travel on public transportation as much as you want AND it gets you into a whole slew of museums and attractions around the city, like Mozart's birthplace, the Baroque Museum and the Hellbrun Zoo (just outside the city). You can get cards that are valid for 1, 2, or 3 days and the cost from 23 Euros to 41 Euros for adults (Depending on the time of the year and number of days).Check at your hotel or some of the smoke shops around town to get one. You can also get a 24-hour bus pass at the tourist information centre (main train station) for 3.30 euros; it is 5.30 euros when bought on the bus. To save don't buy any tickets on the bus. 

For a unique way of traveling around the city, hop on one of the horse-drawn cabs. The drivers will take you where you need to go, as well as give you a bit of a history lesson.