lifting weights does very little in actual fingers strength. there are funny videos around when muscular lifters try climbing, they fail quite fast after couple of moves.

you can train a bit with various squeezy thingies, what helps too is just hanging by hands from some thicker bar (so you cannot grip it comfortably). but by far best training, especially in beginning is just climbing. it takes time for all connecting tissue to build up (tendons, ligament etc. - it's not about muscle for fingers). fingerboard can indeed be dangerous if attempted early - say from 6b-6c in french scale.

much of the art of staying on the wall is about proper balance - this can be trained a bit on slackline for example, but again best training is just climbing often.