I think there's an English-language Barri Gotic walking tour run by the tourist office at 9:30 on Wednesday morning. I really liked that walk, and it's in the same general area of a bunch of your planned sights.
Check Renfe.com to see how much a spur-of-the-moment day-trip to Girona or Figueres is likely to cost you. Buying the tickets day-of will probably mean paying full fare. If you're going to feel it's not worth it, you may as well know that now.
The fast trains (55 min.) to Figueres stop at Figueres-Vilafant, which is quite some distance from the center of the town and the Dali Theatre-Museum. Since vacation time is valuable, you probably will not want to walk. There's a bus timed to meet the fast trains, but if the train is late (mine was), it probably will not wait. Step lively so there will still be a taxi hanging around if you need one. The slower trains to Figueres take at least 1 hr. 41 min. They stop at a station somewhat closer to the Theatre-Museum. They are cheaper but I don't know the fare.
The Dali Theatre-Museum doesn't necessarily sell out every day, but its tickets are timed, so you should not just show up, expecting to buy a ticket and walk right in. You might be sold a ticket for 3 or 5 hours later. The jewelry collection in a separate but nearby building is fabulous. It's covered by the same ticket. I was told one can visit the jewelry collection first if desired. Just don't miss your appointment at the DT-M.
The Dali Theatre-Museum is a lot of fun; it can get crowded when a bus tour arrives. If that happens to you, head upstairs for a while. The bus tours never seem to stay anywhere very long.
Girona has more to see than Figueres (which is rather a one-trick pony). Girona is closer to Barcelona so considerably cheaper to reach. The fast trains take about 38 minutes. The slow trains take as little as 70 minutes, though some take much longer. All the trains stop at the same station, which is a bit of a walk from the historic center. Taxis should be available, and I assume also buses.
I think you'll find round-trip tickets a bit cheaper than two one-ways, but that means pinning down what train you want to return on. Buying a fast-train ticket outbound and a regional ticket for the return would probably be less expensive and allow some flexibility. I like Girona so much that I hate writing this, but some people visit it on the same day as the Dali Theatre-Museum. If you're going to try this, get an early entry time at the museum.
