Barcelona

My two favorite modernista sites not by Gaudi:

• Palau de la Musica Catalana (tour required, available in English, takes less than an hour but best to pre-book)
• Sant Pau (English tour or English audio guide; multi-building site which will have you outdoors for a bit; if you plan to use the audio guide you can just walk right up to the ticket desk. This one isn't yet on the tourist radar; enjoy it while it's not a zoo.)

Some wandering in one or more of the medieval districts is pretty much a must: Barri Gotic, El Born, Raval.

There are lots of good museums, so check your guide book and see what sounds as if it would appeal to you. Be aware that the Picasso Museum is extremely popular and you'll need to buy a timed ticket in advance to avoid a very long wait in line. Unfortunately, I'm not aware of a way to avoid crowded conditions inside the museum. Skip it if you don't much like Picasso.

As for any large city, with only two days you will have to be brutally selective. So, read about the attractions and see what appeals most to you.

I agree with acraven's recommendations for non-Gaudi Modernista sights. I also recommend that, on a short visit, you not try to overdose on Gaudi sights; it would be not only exhausting, but expensive. Pick one or two, and leave time for other things.

My favorite things were the Casa Batllo (acraven didn't like the crowds; I enjoyed it so much I didn't mind them), the Palau de Musica Catalana, the Museu Frederic Mares, and the Museu Nacional d'Art de Catalunya (the fact that your ticket is good for two days is the first clue of how much there is to see here).

Here's Rick's list of what he considers the highlights of Barcelona (scroll down to "At A Glance" and click to open it): https://www.ricksteves.com/europe/spain/barcelona

One thing I didn't care for at all was the Ramblas. You'll see parts of it if you go to the Boqueria Market; that's sufficient. On a short trip, don't waste time there, when there are so many other wonderful things to see.

The Picasso Museum is a must if you are a fan of his work. It was wonderful, I loved it.

A tapa's tour for sure

Get lost wandering around in the Barri Gotic and dropping in a few watering holes. Meet some locals.

Day 1
-Sagrada Familia
-Park Guell
-Passeig de Gracia - Casa Mila & Casa Batllo
-Montjuic - MNAC & Miro

Day 2
-Palau de la Musica Catalana
-Ramblas - Boqueria markets & Placa Real
-Bari Gotico - Cathedral
-El Born - Picasso & Basilica Santa Maria del Mar
-Parc de la Ciutadella & Barceloneta

I'd balance my time between modernista sights (Gaudi or otherwise) and older sights.

So Barri Gotic, El Born and surround, cathedral, Santa Maria Del Mar (my favorite church).

And in the evening be sure to find somewhere great to eat. Barcelona has wonderful food. And drink - cava is the classic but I also noticed a lot of vermut. A food tour would be nice.

La Boqueria market, on Las Ramblas, in the morning for fresh churros and chocolate; the Plaza Real, a charming square, look for the adorable old spice shop; the regular old Cathedral, has great view from the top and a little duck- and plant-filled cloister.

I agree with acraven, and also very much enjoyed both the Palau del la Musica Catalana (I went both for a tour, and for a performance) and the Sant Pau Hospital Complex (I took the English language tour, just three of us plus the guide, and was amazed that this beautiful complex was built as a hospital for the poor of Barcelona.) Sant Pau is just a 15-minute slight;ly uphill walk from Sagrada Familia.
I also very much enjoyed the food tour by Taste Barcelona, which included a historical walk through 4 downtown neighborhoods with 3 or 4 excellent food/beverage stops. Pricey at $125 for 5 hours, but so worth it! Covers a lot of ground and a lot of food, a great introduction to Barcelona and Catalan cuisine.