I love Barcelona, based on my one trip. Apologies for being lazy with accents below, and probably some stuff that's outside your schedule, and the general braindump of it all.
Good advice generally above.
We stayed at a tiny hotel not right at La Rambla, but not too far away. Based on that, I don't think it'd be a terrible walk to a couple of my first recommendations:
I really liked starting the day at a little coffee shop called Faborit (Passeig de Gràcia, 41, 08007 Barcelona; there's a really nice outdoor seating area out back), which has the added bonus of being next door to Gaudi's Casa Batllo, which is less overrun than the Sagrada Familia, and totally worth a tour. Not that the Sagrada Familia isn't worth it, because it totally is. We went to the cathedral near sunset, and I gather it's better do do it early or late, as the sun comes in through differently colored stained glass depending on morning/evening. It really is a stunning place. Casa Batllo was more to my tastes, and had a lot of interesting features that were very different, both in terms of design and construction details...
NOHOME's right about paella. Walking on the beach and stopping for paella is something I would recommend very, very highly. About the only thing I didn't get right there was that I had zero cash on me and wasn't prepared for the otherwise reasonably-casual place to have a bathroom attendant I didn't have a tip for... Again, you're not a miserable walk from there.
I've never been there, but stumbled into some really tasty beer from Barcelona since then, and if you happen to wind up near the brewery... Cervesa Montseny.
It's a neat place to just wander around. From La Rambla you're also right at the Gothic Quarter, which is a good area to do some of that wandering...
We did a tour to Girona with a stop at the Dali museum in Figueres; that was a day well spent. Girona's beautiful (it's also where a lot of Game of Thrones is filmed, so there was more GoT stuff than straight history as I might've preferred on the tour, but it was still awesome). The Dali museum was neat, but I was mostly struck by how the sky in Figueres I'd swear actually looks like the skies in a lot of his paintings. It was really neat.
The hop-on/hop-off bus tour was neat, too, with some guidance, some sights, and a neat loop that took us to the Museu Nacional d'Art de Catalunya and over Montjuic.
Don't miss the jamon Iberico.