I exercise on a stationary bike 3 times a week and then I need to shower afterward. What do I do during the nine days? Do I need to stop exercising?
You can continue exercising. Regarding taking a shower I can’t judge your personal situation but the general guideline are that even if one is uncomfortable, that is not enough of a reason to take a shower. The point of not showering is that we should be uncomfortable as a way of mourning over the Bais Hamikdash. The poskim do say that when a person very uncomfortable, because of the sweat then a cool shower may be taken. Additionally you can wipe yourself around with a wet washcloth. There are times when a person can be lenient, but the has to be a genuine need, and not just when there is an excuse to take a shower.
Halichos Shlomo 14-18, Dvar Halacha 22,27, Divrei Yatziv O:CH 2-237, Shevet Halevi 8-127, The Three weeks. pg. 90 ftnt. 162.