Barcelona Travel by Hop-on-Hop-off Tour Bus
To see the bigger picture of what Barcelona is all about, a bus tour is a good way to start. You can study the guidebook for weeks before you leave, but the bus tour, which stops at all of Barcelona's main sights, is a great way to see what Barcelona has to offer and how far apart everything is. If you start early, you can fit in a quick view of most of the city's sights in one day. The perfect first day in Barcelona.
Barcelona Travel on Foot
A tour bus takes you to all the major stops, but after that, you're on your own. Barcelona's most popular tourist area - Las Ramblas and the Gothic Quarter - are walkable (in fact, much of it is pedestrianized), so your most likely form of transport will be your feet.
So once you've explored Barcelona at the macro level with a tour bus, it is a good idea to get your ground-level bearings the following by taking a Barcelona Walking Tour.
They're cheap and will probably save you money when you later realize you don't need to take taxis so often after all. Alternatively, if walking is too, well, pedestrian for you, try one of these Sightseeing Tours of Barcelona - just about every form of transport imaginable can be used to tour Barcelona.
Barcelona Travel by Metro and Regional Train Services
If you didn't get to see all of Barcelona not-so-central sights on the tour bus, such as La Sagrada Familia and Parc Guell, the best form of transport is the Metro.
Read more about the Barcelona Metro, including information about buying tickets: Barcelona Metro.
Note that there are a couple of regional train services you should be aware of. Read more about Barcelona Trains.
Barcelona Travel by Taxi
Taxis are also an option, but I'd only recommend these when the metro is no longer running, as they are a lot more expensive.
Are Spanish taxi drivers trustworthy? As a driver in Seville once told me, most of them are, but all the bad ones are at the airport. But regardless if where you are, make sure you're charged on the meter. Try asking for a price at the start - they won't give you one, but that will ensure that they use the meter.
If you need to get from the airport to the city (or vice versa) and don't want to risk a taxi (and you'd prefer to avoid the bus), then you could get an airport transfer.