Leipzig's centre is a so-called Umweltzone (environmental zone), meaning you need to obtain an Umweltplakette (emissions sticker) if you brought your own car from abroad. Check www.umwelt-plakette.de for details.
Buses and trams are run by LVB, which operates an information kiosk outside the Hauptbahnhof. The central tram station is here as well. There's a second LVB information kiosk on Markgrafenstrasse. Single tickets cost €1.90 for up to four stops and €2.60 for longer trips; day passes are €7.40. The S-Bahn is run by Deutsche Bahn, but the ticketing system is integrated, so LVB tickets are valid on S-Bahn services within Leipzig.
Taxis are expensive here. The minimum charge is €3.50, and each kilometre €2.50.