Girona has a very atmospheric medieval center with a walkable wall, two historic churches, an art museum, etc. Highly recommended. It does get a lot of tourists, but they are mostly in the northern part of the historic area, so you can spend time nearly alone by wandering elsewhere. Accessible by train from Barcelona and worth an overnight or two, though plenty of people treat it as a day trip.
Figueres has the Dali Theatre-Museum and associated jewelry collection. I'm not a fan of surrealism, but I still enjoyed the Theatre-Museum. I don't think the Theatre-Museum sells out way in advance, but I wouldn't plan to just walk up and buy a ticket; it's definitely a stop for bus tours and tickets are limited, so I think you might end up hearing "come back in four hours". Figueres is beyond Girona; it's accessible by train from Barcelona. Personally, I wouldn't want to combine Figueres and Girona on a day trip from Barcelona, because I think you'd have too little time in Girona, but some folks definitely do it. (Figueres doesn't have a lot of sights besides the Dali Theatre-Museum, and the city just isn't special.)
Cadaques is a picturesque former fishing village--all white cubes dripping with bougainvillea (but I was there in the summer). The bus trip from Figueres has some really nice scenery. Some people head there for the Dali home in nearby Port Lligat (walkable from Figueres in 15-20 minutes), but I just visited the town itself. Note that prior to the pandemic, tickets for the Dali house sold out early, so that would be the first thing to nail down in that area north of Barcelona.
Girona makes a fine base for covering Figueres and Cadaques as well as Girona itself. It is possible to visit Figueres and Cadaques in a single day trip from Girona, but an early start would be prudent. If you decide to do this, I'd head to Girona on your arrival day so you finish up in Barcelona without having to worry about traveling back there the night before your flight. It's inefficient to have two separate hotel stays in the same city; I'd do everything possible to avoid that on a short trip like this.
Also accessible by bus from Girona (but in a different direction) is the small medieval town of Besalu. I believe there is bus service from Barcelona as well (perhaps the same bus). Pre-pandemic there was a one-day bus tour of medieval Catalan villages that included Besalu and some others. I haven't taken it, but it has been recommended on the forum. That tour originates in Barcelona.
Since you mention architecture as an interest, you should research Zaragoza and see whether you think it would appeal. Zaragoza has far fewer tourists that Barcelona. There's an attractive historic center and a restored Moorish palace (the Aljaferia) that's open to the public though used for government offices. Architecturally, it makes an interesting contrast to Barcelona.
There are also attractive coastal villages, but I've only been to Cadaques and Sitges (artsy and with at least two museums); I can't give a good overview of the options in that category.
Although I've spent about twelve days in Barcelona recently, I haven't yet been to Montserrat.
For a lover of modernista architecture, there is a great deal to occupy one's time in Barcelona. In estimating how many days you'll have available to spend outside Barcelona, don't overlook the need to prebook timed-entry tickets for a lot of Barcelona sights. Having locked-in arrival times at a bunch of places makes it impossible to be a maximally efficient sightseer. You need to allow sufficient time to enjoy the first sight and then to reach the second sight. The entry tickets are often quite expensive, so it would be a shame to have to leave a sight before you wanted to.
Edited to add: The Sant Pau modernista site in Barcelona gets few tourists.
