Going to the gym is a great way to get into the habit of taking regular exercise. 90% of the difficulty most people face with attending the gym is
actually turning up in the first place. Once you're there, you just get on with your workout without a second thought. So if you make it part of your routine, say every Tuesday at 6pm you'll get up and head to the gym, it will become a habit and you will start to make steady progress.
If your gym is so close to home, you might want to think about whether it'll be easier for you to get changed at home, ride to the gym, work out... and then ride back home and take your shower there in the privacy of your own bathroom. That can help with some of the locker room dysphoria. I must admit that the locker rooms are the most uncomfortable part of the whole gym experience for me, which is why I use the disabled room instead. That'll change when I'm post-op and feel safe to use the men's.
I wasn't sure from your message whether you meant that they have separate men's and women's exercise areas or whether it was just the locker rooms. If it's exercise areas, you'll be absolutely fine if you pass 100%. Nothing to worry about there at all. Heck, there are a couple of cisguys at my gym who are shorter than me (and that's saying something!); and there are several with larger moobs. It's all good.
Also, I use my iPhone (but any MP3 player will do) to distract me from other people in the gym. Music helps to keep me focussed on what I'm doing, but I also keep notes of my workouts so that I know what I did last time and how difficult I found it. Then I can work towards improving my performance next time. Generally speaking, I slightly increase the difficulty of my workouts once a week, which keeps me on my toes. Also, if I'm doing the 'long, slow burn' (low-speed, low-intensity cardio workouts for a long period, such as slowly riding on a recumbent bike for an hour) I tend to put on a movie or a TV episode so I can zone out of the room. That way, I don't even have to think about what anyone else is doing.
It's natural to be nervous, but you'll be absolutely fine and after just a couple of sessions you'll feel like you've been doing this forever. Like Nike said: "Just do it!"