I've only visited the museum in Leipzig. I'm a huge Bach fan, but I even I found the museum a bit on the dull side (then again, I find that museums seldom do justice to music of any genre. Kind of like hearing food or smelling art). I only learned a little that I didn't know before, and the museum houses very few artefacts that have a historical connection to the man. You could probably spend as little as one hour or as much as three. It's not a very large museum.
The Thomaskirche sits right across from the museum. It isn't the most elaborate church I've ever visited, but if an organist is playing at the time of your visit...well, there's something really uplifting about hearing the music of the master played in the building of its origin.
The Nikolaikirche is more interesting as a physical building. I don't think I've ever seen a church with a similar floral motif in the decor, and definately not typical for a Lutheran house of worship. To fully appreciate this building, make sure you read about it's importance in the fall of communism.
Leipzig is actually one of my favorite cities in Germany, and certainly one of the most under-rated. It doesn't have a plethora of museums, but it's such a beautiful, vibrant, walkable town. And for a city that was once part of the DDR, it retains a surprising amount of its historical buildings (mostly from the Empire period). Dresden might have better museums and the historic core is definately grander... but it seems rather divorced, both spatially and chronologically, from the larger city it occupies.
In Eisenach, if you can't understand German, you will need to time your visit to the Wartburg very carefully. The castle offers English tours infrequently. Oh, and for a classical musician... read up on the Wartburg's connection to the Tannhauser legend.
