Probably the best known area is Las Ramblas, and this is probably the first view of Barcelona that many tourists will have; the Placa de Catalunya at the top of Las Ramblas is where visitors will change buses on the way from the airport, many tour buses also stop here. The train station underneath the Placa is probably the best station for a general sightseeing day.
There are lots of places to eat on Las Ramblas, but if you go a bit further off the beaten track, you will find cheaper, and generally more authentic, food and drink, there are deals to be had on Las Ramblas, but they are always based on two eating, not much good if you are by yourself.
For the less adventurous, you can also find McDonalds, Burger King and Subway along here.
Partway down the road, on the right, you will find the market, La Boqueria, a fantastic place for fresh and homemade produce, unfortunately it has become more touristy in recent years.
You can also find a Wax Museum, a Museum of Erotica, an 'adult' shop and the military police station as you walk along.
At the far end of Las Ramblas, you will find the marina and the Mare Magnum. This is a shopping centre with mostly boutique shops and eating places.
Photo courtesy of Maremagnum.es
From Las Ramblas, Mare Magnum is reached by a wooden bridge, known as La Ramba del Mar which can be opened to let the tall ships which moor in the marina out into the open sea.
Passing through the shopping centre, you will find the IMAX Port Vell cinema and the aquarium.
If you continue this way, you will come back to the main Passeig de Colom. Left will take you back towards Las Ramblas, and right will take you past the Estacio de Franca train station and on to the city park and the zoo.
A general tip when in Barcelona, keep an eye on your bags and purses, personally I have never had a problem with pickpockets, but I know others have.