Names of places to visit are definitely welcome! Especially if it involves food
Great trip! Bit on the shortish side but you'll know that.
I can recommend checking out the district of
Gracia
in Barcelon. Lived there a good while and love it. Small narrow streets with a lot of small shops, cafés, restaurants, nice squares and so on. It's touristic (as the whole city) but not remotely as congested as the areas around ramblas. I'd check out the Eastern area between metro station of Fontana and Joanic. Good tapas can be had at th Placa de la Vila de Gracia. Place is called Nou Candanchu - nothing fancy but nice atmosphere.
Also, if you happen to check out the Ramblas go to the big market and get some food / fruit drinks on the cheap. It's called Boqueria (mercato de..), the restaurants on the back side are said to be good but rather expensive IIRC and still next to a parking lot so I'd avaod that. The whole Gotico district is interesting - but take you're time since it's a maze and you'll get lost. When near the beach (not my thing but I guess people like it lol) and hungry: check out this Tapas bar. It's great. But again, take your time since it's always been extremely crowded when I was there. Really nice atmosphere imo but crowded..

Bar Cal Chusco (in the Carrer l'Almirall Aixada)- The whole district is packed with small restaurants though, all serving mostly good tapas / fish.
I'd say you should try to stay two days in Barcelona. But Lisboa and Madrid are beautifully too so it's really up to your schedule (are you hopping by train/car/flight).