More than any city I've been to in Europe, Barcelona is best enjoyed on foot and on a bike. It's small, but not so small that you can walk around it; it's mostly flat; there's something to see every step of the way; and there's a million bars and cafes to stop as you get tired.
I did a Tapas Walking Tour and a half-day Bike Tour, and gave them both five star reviews. Barcelona is really deep from a cultural viewpoint, it's one of those places where you won't know what you missed unless you did the tours and heard the guide. Music, art, history, wine, food, religion... the Catalan region is a real surprise for first-time visitors. While I was there I also did a day tour to Montserrat, way up on a mountain an hour or so out of town; that was cool too, but I liked being in the city better, on foot and close-up to the people.
The bike tour was especially fun; there were a few older people who were a little concerned that they would fall behind the younger set, but we stopped every few minutes and nobody got too tired. We did see a lot of the city we would not have seen on foot, though. And the guide was hilarious. A real bonus.