You're not going to need a car to get around Barcelona, but you may want to rent one if you plan to see anything else of the region - though note that the coastal roads in summer are a nightmare; stick to buses and trains if that's as far as you're going. Major roads are generally good, and traffic, while a little hectic in the towns, is generally well behaved - though Spain does have one of the highest incidences of traffic accidents in Europe. Driving, even with a full car, will work out expensive: fuel prices are only marginally lower than in Britain and almost double US prices, and in Barcelona at least you'll probably want to pay extra for a hotel with parking (which is notoriously difficult in the city center), or be forced to stay on the outskirts. Vehicle crime is rampant - never leave any thing visible in the car.
Most foreign driving licences are Honored in Spain - including all EU, US and Canadian ones - but an International Driver's Licence (available from recognized driving organizations) is an easy way to set your mind at rest. If you're bringing your own car, you must have a green card from your insurers, and a bail bond or extra coverage for legal costs is also worth having, since if you do have an accident it'll be seen as your fault as a foreigner, regardless of the circumstances. Without a bail bond both you and the car could be locked up pending investigation.
Away from main roads you yield to vehicles approaching from the right, but rules are not too strictly observed anywhere. Remember that you drive on the right in Spain. Speed limits are posted - maximum on urban roads is 60kph, other roads 90kph, motorways 120kph. If you're stopped for any violation, the Spanish police can and usually will levy a stiff on-the-spot fine before letting you go on your way, especially since as a foreigner you're unlikely to want, or be able, to appear in court.
In the event of car trouble, the Reial Autoṃbil Club de Catalunya has links with its European equivalents, and there's a 24-hour telephone line for emergency help and information : tel 900 307 307. Its office address in Barcelona is Avda Diagonal 687 (tel 934 955 000).