In my experience, most local races will bring bikes in on race day. Ironman-branded events and some other larger events will post security for transition areas and expect you to rack your bike overnight.
Short answer: no, your stuff doesn’t go in a bag. It’ll generally just get left at your spot until your race is over. You’ll want to lay out your transition area and practice it. I wrote a really long post about transitions here with a lot of tips built up from 13 years of racing.
I have always done this, just making sure my bag is not reachable from outside the transition area. I usually stash it off to the side, out of the way, inside the transition area near my spot with keys, wallet, etc. not readily accessible. You could also stash your car key inside your tri shorts, assuming it’s not a fob.