I am flying to Berlin Tegel Airport (TXL). I need to get to Leipzig. Is it possible to get a train from the Airport? Or do I have to get a connection train? I read something about Rail& Fly. But I am not really sure about the train from Berlin I will be carrying luggage, which will be relatively heavy (~30KG)
In Berlin, the bigger problem is that there is no long-distance train station in any of the airports. In spite of the official website's rhetoric, Tegel is actually relatively poorly connected to the local and national transport networks, you need to take a bus to get anywhere, there isn't even a good S-Bahn or U-Bahn connection. Since it's much closer to the center of the city than many other airports, it's not that bad but you do need to get at least one bus connection.
Once you get to one of Berlin's major train stations (e.g. the new “Hauptbahnhof”), you can easily catch a direct train to Leipzig, possibly with a Rail&Fly ticket. Rail&Fly fares can only be booked through your airline but if there is a train, you can always take it with a regular train ticket as well, you don't necessarily need Rail&Fly or any other special fare.
Incidentally, Tegel was slated to be closed some time ago but Berlin's new airport (built around the old Schönefeld airport) also had many problems with it's connection to the rail network and isn't even operating yet, several years after the planned opening so Berlin is a very poor choice if you are looking for an easy train connection.
You can reach the "Hauptbahnhof" from Tegel by riding the TXL-Bus (public transportation) which runs every 5 minutes. The fare is about 2.7€ and it will take about 15-20 minutes. Take a direct train to Leipzig from there. This is the Link to the german railway authority called Deutsche Bahn