Going to a gym or taking fitness classes can trigger breakouts but since I have already told you that sweating does not cause acne, what could it be? Well, you don’t necessarily need to wash your face before or after (I know, this completely goes against all of the advice you have ever heard) AS LONG AS you are wearing acne-friendly makeup because if you are wearing makeup that doesn’t contain any comedogenic ingredients then it sits on top of your skin and working out won’t cause it to slip into your pores. If you are wearing pore-clogging makeup, don’t bother washing your face because by the time you get to the gym that makeup has clogged your pores anyways :D You also don’t need to wash your face before or after working out (I know, another piece of advice that completely goes against the advice you have heard all your life) because, again, sweat doesn’t cause acne. Here are the list of things you do need to do though to avoid breaking out when you workout:

Always bring your own towels that have been washed with acne-friendly laundry detergent and not dried with dryer sheets or fabric softener. Gyms and fitness studios often wash their towels with heavily fragranced detergent that can trigger a breakout if it touches your face or body. If you forget a towel, use paper towels or your t-shirt to wipe yourself.
Never use toiletries (shampoo, conditioner, body wash, lotion, etc) provided by a gym or studio unless you have checked the ingredients against the list of comedogenic ingredients. If the ingredients are not listed, do not use them and bring your own toiletries instead.
If you use hand soap or hand sanitizer at a gym/studio, do not touch your face after because it might contain pore-clogging ingredients.
Bring your own mat. “Dirty” mats are not the problem (remember, dirt and bacteria do not cause acne) but if the mat is washed with a cleaner that contains a pore-clogging ingredient, it can trigger a breakout.
If you have to use a mat, equipment or machinery that is not yours (e.g pilates reformers, yoga mats, etc), keep your face off of it or put your own towel on it to protect your face. If the equipment was cleaned by the studio, they might have used a pore-clogging cleaner and if it wasn’t, the person who used it before you might have transferred pore-clogging products from their faces or bodies onto the equipment.
Acne mechanica is caused by pressure and friction on the skin which can irritate acne and cause more inflammation. So try to keep as much pressure and friction off your face by wiping gently with your towel, not touching your face with your hands, and wearing loose clothing if you get body acne.