Karen, I guess you mean Freiburg im Breisgau, there is more than one Freiburg in Germany.
For the Swiss portion, there is no additional discount for purchasing in advance above the 50% you will be getting with the Half Fare Card.
For the German portion (Freiburg im Breisgau to Basel), on an ICE train it takes 47 minutes, and costs €19.90 if you buy in advance on the DB website, €26.60 if you buy on the day. However if you buy a Baden-Württemberg ticket ( bought on the day) it costs €24 plus €6 per extra passenger. This works out cheaper for 2 or more people, but you have to take a regional train and it takes 1h10. Your choice.
The two stations in Basel are Basel Badischer Bahnhof and Basel SBB. The ICE train stops at both, you need to get off at Basel SBB (the second one), buy your Half Price Card and ticket to Mürren, then get on your Swiss train. If you take a regional train, that terminates at Basel Badischer Bahnhof. You need to get another train for the 5 minute trip to Basel SBB (covered by your Baden-Württemberg ticket) from where you can catch your Swiss train.
Hope this answers your questions.