Are you looking for an apt or a hotel??

Plaça de Catalunya. Is the very center of town

There are many threads on Barcelona...just hit search

A budget and your dates would help.

I guess it would. Budget $75-130 night, probably a hotel or B&B would suit us best. I think apartment Neal's are usually more money and by the week rentals. We will need a room for 4 nights.

You might want to check booking.com to see what's in your price range. There are not many hotels in that price range, unless you're considering a one-star property, but again, the dates are very important.

I'd like to take the train from Perpignan to Figueres to check out the museum, then take the trail the rest of the way to Barcelona that evening. I can just buy my ticket at the train station when I am there correct? I do not have to pre purchase or is it better to buy online ahead of time?

Very cheap hotel on Carrer Boqueria called Hotel Condal. Clean, very nice staff, 2 steps away from the Rambla. The only downside is the quality of the beddings.

barcelonagal is right, everything is well-connected by the train system. We did a home exchange last summer for 3 weeks, and were about 20 minutes from the centre, but also had our 4 year old with us. The nice thing is that in Catalonia everyone takes a bit of a siesta, then stays open late, so you don't have to feel bad about going back to the apartment for a midday nap!!