How to beat the heat in summer

Northern Spain, from Galicia to Navarre (Galicia, Asturias, Cantabria, Basque Country are on the coast), is not that hot in the summer and tends to be rainy and wet, even in the summer. Expect temperatures in the 80s as a standard, very occassionally in the 90s. Lots of mountains, green valleys, beautiful landscapes, rough sea, beautiful beaches surrounded by forests...A very different Spain.

Do your touring early in the morning and do the Prado and museums after lunch.
When outside wear a good hat protecting you head and face from the sun. Drink plenty of water so you don't get dehydrated.

Also, agree with previous poster, if you venture north to the Basque Country it will be cooler.
Still, don't miss Toledo and Segovia on day trips from Madrid.

If you had a rental you could visit a few out of the way places. Since I don’t travel to Spain in August, I’m not certain how busy these places get, but I’m certain they’d be less busy than Madrid.

A sample itinerary could include Chinchón, Cuenca, and Teruel. Arrive at Madrid and head to Chinchón that is less than an hour’s drive and spend the night to get over jet lag. Chinchón is small and its draw is the main plaza. Then head to Cuenca (1.5 hours) to spend a night. Cuenca’s old town only takes a few hours to see, so maybe head over to La Ciudad Encantada for a walk. After Cuenca, head to Teruel (2.5 hours) for the remaining time that you have. Teruel is a very nice town that is nice to walk around. From it one can visit Albarracín (40 min), walk through the town and possibly walk the Paseo Fluvial Rio Guadalaviar. It’s a few kilometers path that goes around the town. Another very nice walk closer to Teruel is Camino Natural del Rio Guadalaviar. It’s about a 6 miles round trip walk along the river. Some walkways extend out over the river and there are some pretty steep walls one passes. There are no comfort facilities along this walk so use the restroom before going and have plenty of water. It is in an out of the way location, but only 5 miles from downtown Teruel. Teruel to Valencia is about a 1.45 hours drive.

When we visited Chinchón, Cuenca, and Teruel, we stayed at the Paradors in each place and had a rental car.

I like all the places mentioned by jaimeelsabio and would tend not to go all the way up to the north coast with the limited time you have. (However, I wouldn't plan to be in central Spain in the summer, either; be sure you're prepared for the heat.) Albarracin is a great suggestion if you're going to have a car; it's difficult to access by public transportation. When I made the trip, the bus from Teruel arrived around 4 PM and departed early in the morning, so you'd really need to spend two nights in Albarracin to accomplish much at all.

Teruel itself is on a spur rail line connecting Zaragoza (very, very hot in summer) to Valencia; it's manageable without a car, but a car would be more efficient.

Cuenca is super-easy without a car; it's on the express rail line between Madrid and Valencia, though you'll need some form of transportation from the AVE station into the town itself; there's bus service but I am not sure of its frequency. There are a couple of small art museums in Cuenca that I liked, and it's fun to wander around the medieval center. To me it's worth more than a couple of hours, but I'm a slow traveler.

For what it's worth, the fastest way to get to Valencia from Madrid is by train--that's 2 hours vs. about 4 hours by car. Fares on the fast trains vary with demand; they'll be lower if you buy tickets early

You could reach the Sierra de Gredos or the Sierra de Guadarrama, easily from Madrid.

Parador in Sierra de Gredos, near Avila:

https://paradores.es/en/parador-de-gredos

Wines:
https://spanishwinelover.com/las-pedreras-young-talent-and-new-vineyards-in-the-gredos-mountains

I loved Cuenca, good food town! We drove from MAD, stopping at Segobriga. There are swimming opportunities in the gorge and nearby. I didn’t swim, as I was there in May and did some hiking in the region, but I found this: https://www.cuenqueando.com/en/blog/507/the-best-natural-pools-in-cuenca
Are you looking to skip Madrid entirely, and do you have any time planned in Valencia? You must tell us where the wedding is, lol.

ETA Do your dates coincide with the solar eclipse?

The Parador de Gredos is in a very picturesque area and there are a number of places to go hiking. The Parador itself is in a fairly remote location. I know buses go by it, but don’t know the frequency. We spent a night at it and wished we had stayed an additional night. We visited it after spending a night in Ávila and before spending night in Oropesa. We had a rental car.

Base in GIRONA and escapades to the Catalan Pyrenees....

https://www.girona.cat/turisme/eng/index.php
https://www.visitpirineus.com/en

Enjoy!