As soon as he can start climbing out, you should move him. The timing varies for everyone. My older child moved at 12 months, my 24 month old son is still in his crib. If you have a convertible crib, you can just take off one side and use it as a toddler bed. Otherwise, I would suggest you let him help pick out his new bed, or at least the sheets and blankets. I ended up buying a bed at IKEA because they had one that could be short for a toddler and then grow as they get older. It was also very low to the ground so I didn't worry about him falling out. Be sure to get some bed guards so that you don't worry about him falling when he is in the bed.
As far as the diaper, unless he shows interest in the potty, I would not start training yet. You can put a potty seat in the bathroom and offer it but he might not care yet. I was planning on starting my son agressively around 27 months although he does sit on it occasionally now. My daughter was trained at 2 1/2 years. If he continues to take off his diaper, make sure he is being changed frequently and he is not taking it off because it is dirty and he doesn't like it on. I would also suggest putting him in a onesie so he can't get to it. Good luck.