One of the best ways to explore Barcelona is on a scooter. This is one of the most scooter friendly cities, with easy access to narrow alleyways, small courtyards and plenty of out of the way places you can really only get to on foot.
You'll be able to cover plenty of ground in a short period of time with a scooter, not to mention you can venture up the mountain to Montjuic, Tibidabo and check out Poble Espanyol and Olympic Stadium amongst many other places, all of which are more easily accessible on a bike.
This is a great way to spend a day or even an afternoon, especially during summer when you can enjoy the breeze; which is much needed during the heat.