The old people (me) among you says wear your engagement ring on your left hand. When you are married you take the engagement ring off, place your wedding ring in its place and then return the engagement ring to your left hand. I don't know why? Never thought to ask, I do know that when someone looks to see if your engaged, married or single, they look at your left hand.
Wear your engagement ring on your right ring finger. Tradition holds that this is the closest route to your heart. You remove it on your wedding day, let your groom put your wedding band on your finger, then later, you put your engagement ring back on, or both at the same time. The band always stays first.