From Airport Leipzig-Halle there are two trains you can take to reach Leipzig main station. They leave at 7 and 23 minutes on the hour from the airport. The trip lasts approximately 15 minutes. The price of a ticket is 4.20 € or 6 €. You can purchase tickets directly at the train station or online via bahn.de. Choose "Leipzig/Halle Flughafen" to "Leipzig Hbf".
There are taxi stands in front of Terminal B at the airport and in the city centre of Leipzig, e.g. in front of the main railway station. It takes you about 30 minutes to travel to Leipzig main station. The journey to Leipzig main station from the airport will cost about 30 Euro.
If you plan to use trams or buses every day, you may consider buying a seven-day ticket (21,10 €).