Some lesser known things from personal experience:
Barcelona
Go watch the sunset at Bunkers del Carmel, cerveza in hand
Way more interesting than La Rambla is the alleyways of Gotic and Raval surrounding it. Explore it all there's cool shit everywhere
Valencia
Get horchata at the Mercat Central, make sure to get some fartons as well
Walk/cycle the Jardines de Turia
Eat paella (ok this is well known but please don't eat paella in Barcelona and Sevilla when you're coming here)! El Raco de Turia is a great option. Try actual Valencian paella with chicken, rabbit, snails
If you can, go to the Albufera. Hike through rice fields etc. it's pretty cool
Holy shit I just read you're going there next week. I'm very jealous you're there for Las Fallas
Sevilla:
If you want a very local Spanish market experience, go to Mercado de Triana
Sevilla IMO has the most lively tapas culture of the 3 places you're going by far, also I just love all the tostadas for breakfast
Plaza de Espana is absolutely amazing but don't sleep on the nearby Parque de Maria Luisa. Such a diverse bird population and all kinds of water features.