When we moved the kids into the toddler bed I had the crib sheet on the mattress and had made some flat sheets from a twin cut in half. I tucked them in and put the larger crib blankets on top. The kids are 11, 9, 6, 3, 2, and 1 now an NONE of them use anything but a blanket or sheet they favor. They kick off everything and only use what they are comfortable with, It just confuses me, I always liked the feel of the sheets tucked in and not binding me up during the night and all my covers neat and easy to manage in my sleep. Kids will sleep however they like.
If you want him to go into a twin, or bigger, then do it when you think it's time. My family used the baby bed until about 15 months, then the toddler bed until they were climbing out and sleeping on the floor to get more room, usually by 3. I have been known to have a crib and 2 toddler beds or a Queen, a toddler bed, and a crib in the same room for kids. My neice moved her son from baby bed to a queen at 18 months, she said why mess with all the other sizes, he'll need a big bed when he's bigger so I am buying it now.