There are great destinations stretched all across northern Spain from Barcelona to Santiago de Compostela. I recommend a detailed guidebook covering such places as Olite, Oviedo, Burgos, Leon and Santiago de Compostela, plus natural areas like the Pyrenees and Picos de Europa.
With the time you have available, you'll have to be selective, and I'd caution you to do some research on actual, day-by-day, historical weather statistics (timeanddate.com) for the last 3 to 5 years so you'll have an idea of what to expect. Aside from Barcelona and surroundings, northern Spain tends to be distinctly cool as well as wet. For me, it's a great area to visit in the summer when most of the rest of Spain is likely to be seriously hot. It's not somewhere I'd go in April, and I am doubtful about May, especially early May--though I haven't checked the weather data carefully.
Barcelona is one of my favorite cities. Like you, I really enjoy walking around and seeing the local architecture, and Barcelona has lots of modernista buildings to appreciate--not just the Gaudi sites everyone wants to go inside. Be sure you have a local map that marks them. As of my 2016 trip the one-euro map sold by the tourist office was very helpful in plotting walks to take me past many of the modernista landmarks. If you're serious about tracking down most of them, the Ruta del Modernisme website will be very helpful. That organization publishes an excellent guidebook, but (though paper-bound) it is very heavy. There's an accompanying coupon book that confers discounts; it will save you money at most of the less-trafficked modernista sites but probably will not be helpful at places like Casa Mila and Casa Battlo. To use a coupon you have to (or used to have to) buy your ticket at the site itself, whereas a sane visitor who doesn't want to spend hours in line should buy online tickets for the most popular Barcelona sites in advance.
I'd recommend staying in Barcelona rather than commuting in every day. You will find crowds near (and also inside) the Gaudi sites, but if you stay in an area like the Eixample and not right along the Passeig de Gracia, you'll find the sidewalks peopled mainly by locals rather than tourists. You'll know you're in a big city, but the feel will be neighborhood-y. The subway system is excellent for longer trips across the city, but I prefer to walk whenever possible, because the architecture is so nice.
The best way for those who dislike crowds to enjoy Barcelona is to include a lot of non-Gaudi sites in their sightseeing. I especially recommend the Sant Pau modernista site, which as of 2019 still had not been discovered by the tourist mobs. It's a lovely, multi-building complex not very far from La Sagrada Familia.
The historic city of Girona, north of Barcelona, has a huge medieval district. It's popular but not as over-visited as Barcelona.
In the Basque Country I prefer the less-touristy Bilbao to the stunningly-situated San Sebastian. The vast majority of the tourists in Bilbao don't seem to go far from the Guggenheim Museum. The Casco Viejo (historic center) is very atmospheric and seems to be visited primarily by locals. In San Sebastian the smaller historic center has been turned into one tapas bar after another.
I adore Toledo for its huge medieval district. Madrid, to me, is more about art museums than architecture, so you might trim your time there to allow for a good visit to Toledo. There are some buildings in Toledo with Mudejar (Moorish) interiors. Since your first trip will not include Andalucia, it would be great to see some Mudejar architecture in Toledo.
Between Madrid and Barcelona is the major city of Zaragoza, a stop on the AVE rail line. It has a magnificent Moorish palace and an attractive historic district.