Madison has been on a toddler bed for about 3 months. We just took the mattress from her crib and the same sheets and put it in the toddler bed. She hates sheets and blankets on her and doesn't like pillows so it looks basically just her her crib did only without the rails. I didn't buy any toddler bedding cause I know she hates blankets so it seemed like a waste. We just use her crib sheets, but I let her pick out which one she wants on the bed everytime we change the sheets, which I never did when she was in the crib.
If you haven't done it already, I would keep PJ on the same mattress and give Brian the new one since PJ is probably already used to the feel of the mattress from his crib so it might make the transition easier. I know it sounds like it wouldn't matter, but my SIL put the new mattress on the toddler bed and her daughter would not sleep in it. They decided to switch the mattresses just on the weird chance that she like dthe old mattress and she no longer had a problem with the bed. They realized afterwards that the new one was a little firmer (I guess since it was new) and also made a sorta crinkly sound when she climbed around on it and the old mattress didn't do that. Unless of course you bought a toddler mattress for the toddler bed, then you can't do that cause I think those are too soft to be used as crib mattresses.