Rick's guidebooks are usually very helpful for saving money. He'll help you avoid costly mistakes.
The Ramblas is, to me, just a street--albeit one with a lot of people walking along it. Unfortunately, some of those folks are highly skilled pickpockets. Barcelona has more than its share and the Ramblas is a happy hunting ground. If you wander into crowds, especially on your first day when you may not be mentally sharp, be very careful of your belongings. If you're male, don't make the mistake of assuming a front pocket is a safe place for your wallet.
Barcelona has an exceptionally high percentage of sights open 7 days a week. In addition to the ones already mentioned, you have: Casa Mila/La Pedrera, Parc Guell, Sant Pau Recinte Modernista and the Picasso Museum. A lot of the sights are open into the evening as well, so you can stretch your sightseeing days if you need to.
Of the sights mentioned so far, all except the San Pau modernista site really call for buying tickets in advance. Probably not far in advance--I don't think we've had reports of sell-outs weeks in advance. (You can check the ticket situation on each websites by pretending to buy a ticket for the immediate future.) The problem is the length of the ticket lines. You do not want to spend an hour, or more in the ticket line, and quite likely be sold a ticket for a few hours later. That makes for extremely inefficient sightseeing. It's helpful for scheduling purposes to start each day you'll spend in town with an early entry time at one of the very popular sights, and those early time slots seem to be the first to sell out.
Taking one of the slower trains to Figueres station may be the right thing for you to do, because you'll be able to wait to buy the ticket until the day of travel if you want to; tickets on the fast trains would be a lot more expensive if bought on the day of the trip. However, there's quite a time difference. The fast trains to Figueres Vilafant take about 55 minutes; the slow trains to Figueres station take at least 101 minutes. Both leave you with a walk to the Dali Theatre-Museum if you don't catch the city bus. I think its schedule is aligned with the fast trains, at least, but if your train is late the bus won't hang around, waiting. The difference in walking time to the DTM from the two stations is probably not much more than 10 minutes.
I'm a big fan of Girona and have found on two occasions that a full day there wasn't enough for me to fully explore the medieval center. If I were going to try to see both Girona and the DTM on a single day (which I wouldn't do, but many travelers do that), I wouldn't want to spend 3 hr. 20 min. (or more) sitting on trains. As in Figueres, when you step off the train in Girona, you have something of a walk to get to where you want to be, so the travel time is actually longer than indicated by the train schedule.