I plan on travelling throughout Spain in the near future, including areas where Catalan, Basque and Gallego are official languages. I speak Spanish fluently, albeit with an accent, but I've never been to the "bilingual" regions. My question is: is it OK to address people in Spanish or is that considered rude? I know that everyone speaks Spanish in these areas but I don't know if some people will take offense if you address them in Spanish without first saying a few words of the local language. I'd very much appreciate any insight you can give me as I'm loath to commit any cultural faux pas while I'm travelling through Spain!

It's totally OK to speak Spanish in every place.
Catalonia is the most likely place to be confusing, everyone will talk to you in catalan, but not being Spanish, no one will find it rude if you speak Spanish.
In fact, catalans are good businessmen and they care a lot about tourism, it's the most touristy place of the three, so they will be very helpful indeed.

Addressing someone in Spanish in those areas is, definitely, not rude. Spanish is their official language, too. Chances someone can feel offended are less than 1% if you were a Spaniard, but being a foreigner, chances are even lesser (people tend to be more understanding, especially in language terms).

Said that, if you get to learn a few words in the local languages, your effort will be truly appreciated.