Transition to toddler bed


I'm at a loss. My not quite 13 month old climbed out of his crib at the lowest setting two nights in a row. I was really hoping the first time was a fluke but no such luck. Now that he knows he can do it I know he'll keep doing it, so it's time to transition to a toddler bed so he'll be safe (he's totally fine btw). I'm a FTM and don't know how that works though. I know I need to baby-proof the room completely, wall mount the furniture, cover outlets, etc, but how do I make sure he actually goes to sleep? Will he just eventually fall over from exhaustion? He was beautifully sleep trained but the last two nights he just hasn't wanted to go down (hence climbing out of the crib) and both times I've ended up taking him to my bed until he goes to sleep. What do I do??