Barcelona sights for one not terribly interested in Modern art

Spend some time in the Barrio Gotico:

https://theculturetrip.com/europe/spain/articles/the-top-10-things-to-do-and-see-in-barrio-g-tico-barcelona/

Eat

Spend LOTS of time sitting with Sangria and people watching!

Eat

Park Guell is pretty cool no matter your art style

Eat

We found Barcelona to be a pretty inexpensive place to eat & drink

... you don't say what you're interested in... so instead of shooting in the dark.... http://www.barcelonaturisme.com/wv3/en/

The Museu Nacional d'Art de Catalunya is a very good museum. In addition to earlier paintings (and probably sculpture, but I don't remember), it has wonderful frescoes rescued from churches in the Pyrenees, plus other architectural bits and pieces. There's also a good modernisme collection (primarily jewelry and furniture), but that may not appeal to you. I visited the museum twice for a total of about 5 hours.

If the weather is suitable, perhaps a day-trip to Montserrat?

I loved Girona, which has a walkable wall, a good medium-sized museum and a couple of interesting cathedrals, both with English audio guides. The historic district is quite large and great for wandering. Parts of it have lots of shops and restaurants, but some areas are quiet and feel very local.

Barcelona Sights

In Five Days - without visits to key Modernist architecture & art museum sights such as...
•Hospital Sant Pau
•Palau de la Musica Catalana
•Casa Mila
•Casa Batllo
•Picasso Museum
•Miro Museum

Friday
•Ramblas - Boqueria Markets, Placa Real, Maritime Museum
•Bari Gotico - Cathedral, Barcelona History Museum, Museu Frederic Mares

Saturday
•Pedralbes Monastery
•Montjuic - MNAC

Sunday
•El Born - Basilica Santa Maria del Mar
•Parc de la Ciutadella - Arc de Triomf
•Barceloneta - Museum of Catalan History

Monday
•Sagrada Familia
•Park Guell
•Passeig de Gracia

Tuesday
•Montserrat

There is still a lot you can do and have gotten some good examples. The cathedral has a museum and is a very nice Gothic design. A stroll along the boardwalk is nice too.

Montserrat is a popular daytrip as noted, especially if you like some hiking and nature. It's a beautiful place on a fine day although the architecture is a late 19th century recreation of the original.

Nearby in Tarragona are a number of excellent Roman ruins and a fine museum. Makes for an easy daytrip and you can stop at beach towns along the way back, like Sitges. Both are right along the regional train line.

We had the most fun last month with this: http://gocarbarcelona.com/
I have spent 3 trips/ 14 days in Barcelona.
4 hours in one afternoon was lots of fun- we chose a Saturday for less traffic. The cars are reversed tricycles and go up to 40 mph- easy to drive fully automatic-- and the GPS talks to you. We did the " Gaudi Tour" . the basic rate was about 90 euros for 2 people . we added insurance and an extra hour for 10 euros each part-- saw parts of Barcelona I would never have seen- walked back thru the Park and thru the Gotic and subway back to hotel. that's not much more than the hot hop -on/ hop off bus, and we went to lovely parks above the city.

You don't need to be interested in Picasso or Modernism to enjoy this city. Honestly just walking around in the different neighborhoods will give you a great feel for the city, especially if your on a budget getting away from the touristy areas will save you tons of money. Go to El Raval and hang out at a cafe. Get lost in the gothic quarter and eat some churros continue walking through El Born. Explore the different parks such as a parc de la Ciutadella or the Olympic park with the old stadiums and beautiful gardens. If you enjoy architecture you can see other less famous Gaudi buildings such as Torre Bellesguard or Casa Vicens which are closer to Parc Guell. Just enjoy the city by walking around getting some tapas and sangria sitting outside, choose the key things you want to see and then plan your day around that area of the city.