If you're eating in restaurants, stay away from the ones with pictures on the menus. Look for ones that offer a prix fixe lunch (called Menu del Dia, typically around 1-3pm), usually €10-20 for a multicourse meal including wine.
For even cheaper eats, basically every bakery you pass will have premade sandwiches (bocadillos) or empanadillas for a couple of euros.
Be aware that a lot of cafes have three prices: counter, table and terrace. If you want to sit and people-watch, the terrace price is worth it, but if you just want a cheap coffee and pastry, go to the counter.
In Madrid, go to Plaza Mayor on Sunday and walk around El Rastro flea market. Definitely spend time in El Retiro.
Do not eat paella in Madrid if you're going to Valencia anyways. If you're looking for a daytrip from Valencia, the old Roman capital of Hispania, Tarragona, is about 2 hours north by train and has some really amazing ruins.
It's not cheap, but Alhambra is an amazing place and definitely worth a visit. (Fun fact, al means the in Arabic, so Alhambra is "the red one". In Spanish, it's often "La Alhambra", or "the the red one", and while I was in Grenada I heard multiple people say "The La Alhambra".) I would put Alhambra above Segovia on my list any day.
I'm going to Barcelona and Donostia this summer and did research on eSIM cards for local data. I used to travel with a dumbphone and buy a SIM at the airport, so that looks like it's going to be way easier, although tbh I haven't tried it myself.