'put on the suits' in this case means to get dressed in their ironman and captain america costumes.
Boxing matches are in 'rounds' - people box for a set period of time in each round. A number of rounds make up the match. "go a few rounds" doesn't mean to box, but to fight for a while to see who wins.
He was suggesting they both put on all their weapons and armour, fight and see who is the better. I think this was from the civil war film? If so, the fight would also resolve who has the right opinion.