Bus: All of Central New York is served by the Centro bus service. Most areas within the city limits are reachable by bus, but once outside the city it is harder to catch a bus. Service is general very limited and slow. I would recommend renting a car to get around town, if you don't have one already.
Most car rental businesses are located near the airport.
Driving: The easiest and quickest way to see Syracuse is by driving. Any suburb or city location can be reached within minutes by car. Traffic is not bad at all, and the highways connect the major areas easily. For instance, you can be north of the city near route 31 and then drive to the southwestern suburbs within 30 minutes without any problems. Parking in downtown can be tricky, especially during the week. There is parking allowed on the street, which is the cheapest way to park. But, there are also numerous lots and garages to choose from too.
Walking: Downtown Syracuse is rather small. Walking from place to place in downtown is probably the best way to see downtown so you don't need to deal with parking. There are a couple museums in town, some stores to shop in, and plenty of restaurants. The nightlife is centered around Armory Square on the west side of downtown, which features numerous bars and clubs catering to all adult ages. If you're a younger, college student person, a walk to Syracuse University might be more appealing. A walk from downtown to Syracuse University is not far, although it does pass through a sketchy part of town. Don't go alone. Once on the SU hill, look for Marshall Street and you'll be sure to have a good time.