The Coliseum is a large arena to the right of the Town Square in Lamplight Town. There is a healstone in this square, as well as one inside the Coliseum.
Inside the Coliseum there is a NPC named Clankboot. He will set you up to fight a Computer AI Wizard, and a new challenge will unlock every five levels your wizard levels up.
Another NPC, the Arena Caller, will set you up to fight other real-life wizards that are also looking for a challenge. You will have your wizard and one pet. (Decided by which pet has the red circle around it, changed by clicking the helmet and switch mark in the pets menu.)
Finally, there is the Arena Merchant, who will sell you items if you have enough arena points to unlock them.