If you are visiting during University term time then look out for the Uni a capella groups. They perform regularly all over town and there should be posters in colleges and the Covered Market.
The best are Out of the Blue. Awesome covers of modern rock and pop plus barbershop and some classical.
If you get to go it will make your trip!
Dress Code: Come as you are
Classical concerts are held regularly in the Sheldonian theatre and are usually advertised by posters around town well in advance. I've only been to one - Mahler's Resurrection - and it was well worth the five pounds. My only complaint is about the seating - after half an hour in cramped seating you will be longing to walk stretch your legs.
Dress Code: Reasonably smart!