Every class is trained in unarmed attacks at level 1.
Yeah, they even rank up when your simple weapon proficiency does automatically.
Even the wizard!
Heck, considering the basic unarmed attack is finesse and a lot of wizards put something into Dex, they can be surprisingly good at tagging people in the face even if it hurts less than the barbarian doing it. Bonus points if you do it a high level; even with your relatively pathetic AC, attacks, and HP the wizard will absolutely own most if not all of the low level NPCs in the world. I'm suddenly visualizing an archmage who likes to go into random bars and start brawls for sport by screaming "Come and have a go if you think you're hard enough!". Now THERE'S an interesting backstory for how your wizard PC met their master :).