My son was climbing out of his cot from a VERY early age. We did everything possible to try to keep him in his cot but unfortunately we were concerned he would have a fall in the middle of the night (although he never did, he is a little monkey lol) so we made a big deal out of buying his new “big boy” bed. He came to the shop when we picked it out and he tried them all out by jumping and sleeping on them all of cause lol. We didn’t buy a toddler bed because I didn’t think he would get enough use out of it, so we went for a low to the ground normal single bed and bought a side safety rail. We made a huge deal out of the bed that “he picked out” and he picked out a new buzz lightyear pillow and blanket. Then when we got home he of cause helped daddy put the bed together (he thought this was the best game ever, he built his own bed) He absolutely loved his new bed, we were very lucky. And he enjoyed it more then his cot. He never bothered trying to climb out of it, as it was open so there was no point lol and we all got a lot more sleep
I think if you make it a really fun experience then it makes the transition easy. Good luck and congratulations on your new baby!!