Sant Antoni and Poble Sec are a must if you're a foodie. The two neighbourhoods are next to each other, divided by the Paral·lel, but they are very different from each other.
Poble Sec has that blue-collar neighborhood feeling to it, with many inexpensive tapas bars and places to go for a cheap beer. It's great for barhopping and it has a very relaxed feeling to it.
Sant Antoni is the hip, up-and-coming area. Walk along the Carrer Parlament and you'll find a lot of very interesting bars and restaurants. Healthy food, amazing donuts, cute cafés, adorable bakeries... I'm sure you can spend a whole day here, just eating and drinking!
Here are a few of my favorites: