I want to have 2 full days in Madrid and a day trip to Segovia to see the Roman ruins. Do you have any recommendations for Madrid?
You only need half a day in Segovia as it's tiny! If I were you I'd take advantage of walking around the city (Madrid), there's no need to take a bus or metro etc you will see a lot more that way.
Prado Museum is cool for old art, Reina Sofia has the guernica - which is amazing irl (you can enter them for free at certain times everyday, so check before).
You don't necessarily need to visit the Palace but just walk around the grounds, it's most beautiful at sunset. You can eat incredibly well for very little, avoid touristy places! It's a really vibrant city, full of life and lots of things to do and see. You will also find that a lot of people don't speak English, so bear that in mind!
If you do book, I'll give you restaurant recommendations and bars etc. I literally left this year so everything is fresh in my memory and I love it so much!