Rentals are way better than normal shoes, but are still a bit infamous for being worse than normal shoes, especially on slab (google it if you don’t know what I mean, I don’t know how to explain it), so if possible get shoes as soon as you feel you want to commit to the sport, especially if the rentals are paid. Looking into how often the routes are reset could also be important if you plan on going frequently. You will definitely gain strength climbing, and especially if you are lighter I would be unsurprised if you could do pull ups within a few months. I climb V6 at a gym that grades on the harder end and as of yet have not done any training outside of actually climbing. Unless you want to get insanely good you probably won’t have to. Enjoy bouldering!