My kids (twins) moved to toddler beds just after they turned 3. I was fortunate in that they weren't really climbers. One of them figured out how to climb out of his crib just a week or so before we had planned on giving them toddler beds (for Christmas). I don't recall when we gave them pillows. It was much earlier and they started with pillow pets.
If you are worried about him falling out of the toddler bed, you can just put the mattress on the floor. My kids' room has a hardwood floor so we put some of those foam play mats down in case they fell out of bed....which they did the first day they got in it and played around. Later, we just ditched the toddler bed and went right to a twin mattress on the floor.
If you are worried about night time exploration, you can put on one of those safety doorknob covers on the inside of his bedroom door if you tend to close it. Or if you like to leave his door open, put a baby gate up for his room. Either one will also help for nap times (if he still naps). My kids pretty much gave up the nap once they moved to toddler beds.
I have a safety gate that blocks the way to the front door and also have a safety door knob cover on the front door. Other rooms are either closed with a safety knob on or have the risky cabinets safety locked. One of mine knows how to open the safety gate, but to my surprise, he rarely opens it when he's not supposed to. But that's why we have the extra security of the safety cover on the front door knob. We duct taped the doorknob cover on in case they figured out how to get it open so they can't escape out the front door. If you have a sliding glass door, they make safety locks for that as well.
As someone else mentioned, I did find that my kids tended to wander into my room over other places in the house. Even now that they can use the bathroom on their own, they first come into my room to tell me they have to go potty and then go potty by themselves.