Upvote:10
There is no train station at Tegel airport. You best shot is to take the TXL Jet Express bus to the central station (Hauptbahnhof).
There are plenty of train connections to Halle. Every two hours there is a non stop high speed train that does the trip in a little over an hour. However it's pricey: 47 Euro (unless you book a few weeks in advance). There are many other trains with connections which are as little as 20 Euro.
Cheapest would be a bus like this which can be as little as 7 Euro. However, you need to check the departure point for the buses. This one leaves from Funkturm so you need to find the right bus from Tegel to get there.