There is a good selection of tennis clubs in Bern. See the Bern Tennis list here.

At the posh end of the social tennis spectrum, there are three clubs next to each other near the Tierpark, at the back of Jubiläumsstrasse. The main one is TC Dahlhölzli, which has a good reputation, the best teams and some very good coaches. The smaller TC Sporting Club Bern and TC Rotweiss are next door. TC Neufeld, which is up near the Gymnasium Neufeld, is less posh, but friendly and a good place for serious amateur players. There are also lots of other good clubs (see link above).

For serious tennis lessons and coaching, I can recommend TC Dahlhölzli, and particularly Andi Strasser, who coached our son for a while. But I wouldn't recommend Michel Kratochvil, who runs the TC Flamingo in Ostermundigen and TC Sagi in Bolligen. Kratochvil was a good tennis player (ATP 35), but he is now trying to cash in on his reputation as a player. Don't be taken in by the Kratochvil PR. Michel is not a qualified coach, and he runs a chaotic Tennis Academy with a very high turnover of staff and pupils. He is also very expensive (lots of hidden costs).

However, there's no getting away from it: tennis is a very expensive sport here, particularly coaching. Our son plays seriously, and the costs are roughly three times what they were in the UK. Incidentally, when he left the Kratochvil Tennis Academy he went to Swiss Tennis in Biel, which I can highly recommend for any budding Roger Federer.