3 Days in Madrid & 3 Days in Barcelona

First, don't be nervous. Both cities are friendly, just be careful with personal belongings because there are pickpockets. They won't mug you but if there's an opportunity to snatch something . . .

You're close to the San Miguel market in Madrid. It's a good place to graze for lunch or an early dinner ("real" dinner in Spain is not before 9 pm). Take a "free" walking tour on your first day to help get oriented and also to learn about places you may want to visit. Quiz the guide for more ideas. And surf the net - Lonely Planet has great guide books, but also lots of info on their website.

The best place for info on Barcelona is the official tourist site. There's a lot to choose from - try a tapas tour, a guitar concert, and walking tours. From Plaça Catalunya you can walk to many sights but take the metro too - it will save you time. There's a big department store there, El Corte Inglés, with a supermarket downstairs. There are tons of ready-to-eat food.

Take a day trip from Madrid to Toledo and/or Segovia.
A day trip to Monserrat would be nice.

Pickpockets are terrible, especially in Barcelona. Buy a Pack Safe purse and wrap it under your arm and around your shoulder.

The Rick Steves website offers lots of information for free. For example, if you click on "Watch, Read, Listen," then under " Listen" click "Audio Europe App", it will tell you about the Rick Steves Audio Europe App you can download.

Once you have the app, go to Add Tours and Interviews, then go to Spain, and pick which ones you are interested in. This will place them on your Playlist. By listening to these podcasts before you leave, you can a great deal of background knowledge of what you will be seeing, and become more comfortable with the areas before you leave.

For example:
In Barcelona, there's "Barcelona City Walk," "Barcelona," "Barcelona's Xiample Neighborhood Walk," and Barcelona's Sagrada Familia."

In Madrid, there's "Madrid City Walk," "Madrid and Day Trips,"and "Madrid Side Trips"

And lots of other information about Spain: food, music, other areas, etc.

The city walks are taken from his guide books and he narrates them with someone else, and they keep it light with occasional some corny jokes. The others are done with local guides who know the subjects very well.

I listen to these on my walks in the morning ahead of traveling, and it always whets my appetite for what I'll be experiencing, and gives me new ideas.

If you are interested in private or small group tours, we used Forever Barcelona for a 1/2 day walking tour of historical Barcelona. We also did a full day Gaudi tour with them that was very well planned and arranged where we visited all the Gaudi sights in a day. Maria the owner was wonderful to work with. I would also suggest a Tapas tour - we did one with Taste Barcelona (not to be confused with Barcelona Taste). It was only 6 of us and a great way to see some other Barcelona neighborhoods as well as indulge in lots of Tapas and wine.

In Madrid we used Context tours and did small group tour of both Toledo and Segovia. Their private tours are pricey but their small group tours we have found very economical. Their guides are outstanding. Of course you can easily spend a full day or more in Madrid at the art museums.

In Madrid there is the Prado Museum to visit. There is a great park called Retiro which is a very nice place to stroll around. Puerto del Sol is a people watching place for sure. The Palace is an OK place to visit, but not outstanding. Be sure to visit some tapas bars. You can check Yelp or Trip Advisor, etc. and select something close to where you are staying or ask at your hotel which is the best. Try the churros at Chocolateria San Gines near your hotel.

In Barcelona there is Sagrada Familia to see. Park Guell is very interesting to see. Stroll down La Rambla and stop in at the La Boqueria market. At the end of La Rambla is the waterfront.

Google top things to see in....." " and read what may interest you. Enjoy.

You have chosen two excellent cities offering a vast menu of opportunities to fulfill the items you enjoy. Truly, both cities are safe and offer excellent subway systems. Yes, be wise about how to diminish the potential of being "pick pocketed" (we take the simple preventive steps everywhere we travel; including within the USA).
Chani is an experienced traveler whom I believe has been solo on a number of journeys, perhaps you should PM her to help allay concerns.
I a confident your investment into the vast amount of available information defining each cities opportunities will suffice for understanding you shall create a wonderful memorable journey.
Do take advantage of online ticket reservations for popular destinations and strive to "do less" as your walkabouts will help you discover "more"".

The Prado in Madrid is great, certainly. Also in Madrid, there is the Reina Sofia. If you’re interested in modern art, you don’t want to miss it - they have lots of Picasso and Joan Miró, among others.

In Madrid we also enjoyed the Thyssen Museum which includes masterpieces from a private collection.
Of the two cities, be more alert about your belongings in Barcelona which is notorious for pickpockets.

These Barcelona sights require timed entry tickets. In the past some have sold out, so you should not just walk up, expecting to get inside within a few minutes. Even if the day isn't a total sell-out, you might be sold a ticket for 4 or 5 hours later. Your visit to Barcelona is very short, considering all there is to do in the city; you won't want to waste time making two trips to the top sights.

La Sagrada Familia
Parc Guell (no tickets sold at the park)
Casa Mila/La Pedrera
Casa Batllo
Picasso Museum
Palau de la Musica Catalana

The MNAC (large art museum) and the Miro Museum, both on Montjuic, typically do not have significant ticket lines. The Sant Pau modernista site is one where you can normally walk up, buy your ticket, and be inside in a couple of minutes.

I especially liked the Barri Gotic tour offered by the tourist office.

The location of your Barcelona hotel is very convenient. You're within walking distance of the Barri Gotic (in the direction of the waterfront); the Picasso Museum and the Palau de la Musica Catalana are nearby. If you walk the other way, you're not all that far from Casa Mila and Casa Batllo. La Sagrada Familia is about 1-1/2 miles away, so you might prefer to take public transportation over there. Sant Pau is beyond La Sagrada Familia.

I will add to many of the excellent suggestions.
We enjoyed walking tours as our introduction to both Barcelona & Madrid. We used Runner Bean in Barcelona. The guide was very knowledgeable and was helpful in planning our days. We took the Gothic Tour & Gaudi Tour. https://runnerbeantours.com/

In Madrid we used Ogo Tours. They were recommended by Runner Bean. Also excellent.
Both free tours were 2.5 hours and were well worth our time.

As mentioned up thread we also enjoyed the museums and just walking all the areas. We especially liked El Corte Inglés Cataluña rooftop restaurant for their specialties and beautiful view of Barcelona.

Take in a Flamenco show in Madrid. Get lost wandering the Barri Gòtic in Barcelona.

In Barcelona you want to visit the Picasso Museum and the Sagrada Familia. You’ll need to purchase tickets for the latter before leaving home. I would also wander around the Barri Gotic (old town) because there is so much to see.
Consider taking a day trip to Figueres and visit the Dali Theatre-Museum. You’ll need to reserve tickets for this too.
In Madrid visit the Royal Palace, Prado Museum and you want to see Picasso’s Guernica at the Centro de Arte Reina Sofia. Make sure you spend time observing how he pieced it together.
A good day trip option from Madrid is Toledo that will take 45-minutes by direct train.

I have only been to Barcelona once but have spent a fair amount of time in Madrid. You're staying in a great location making plenty of sites walkable. As far as Madrid goes I'd suggest a walk to the west side to take in an art gallery at least once while you are there. The Thyssen and the Prado are both excellent art galleries and I always enjoy the Lavapies neighborhood on the walk back. The Royal Palace would be a good place to start another day as well. Your hotel is very close to the restaurant Sobrino de Botin so I'd be sure to take in a meal in that historic place. The San Miguel market is also worth (or was Pre COVID) visiting and it is super close to your hotel. If you are able to take in a late night Flamenco show you should consider Las Carboneras. We really enjoyed the show there the last time we were in Madrid. Be sure to go online and ensure you have the necessary reservations since you are on a tight schedule. Good luck and happy travels.