You can use ÖBB trains [http://www.oebb.at/pv/en/], they provide several direct connections between Salzburg and Zell am See. Price as at Feb 2012 is €13.50 one way from Salzburg. Online, prices can be quoted as €20, but tickets at Salzburg train station can cost €13.50. Trains are every hour and takes about 90 minutes.
Of course you can use your car. However, in the city you have to pay for parking. From Salzburg eg Airport you can also take a taxi-more price competitive than you'd think. Several companies have 8 seater people carriers and will charge in the region of €120 (Feb 2012) each way regardless of how many people there are.