How do you train to climb without a gym?
If you don't have a gym, there are still plenty of exercises that you can do that can make you a stronger climber.
Depending on where you live, your best bet may be to just find a playground, they usually have pull-up bars and other equipment that you can do some body-weight exercises.
You can do pull-ups and all of its many variations. Push-ups and all of its variations. Running, sprinting, or jogging for legs. You can also do squats and calf raises, which are great.
You can also either buy or make a hangboard and a pinch block. These are the two classic climbing exercise equipment. The pinch block would be the easier to make, or the cheaper to buy, it's really just a block of wood with a hole in the middle of it to attach a rope and weights to it. The only issue is that you need weights for this to be effective, or something heavy that you can attach to it, like a heavy rock.
Other than that, if you can get a hangboard, that will be your next best option. Hangboards are used for training your fingers by hanging from them from your fingertips. Be warned though that it is very easy to injure yourself on hangboards, and most suggest only doing it after 1 year of climbing.
I always go for body weight exercises. There are always exercises that you can do to challenge yourself and that you can find hard, like learning to do handstands and doing handstand push-ups.