Another option with probably some snow but nothing that you can't resolve with good boots, gaiters and poles: Guadarrama.
It's near Madrid where you can "practice speaking spanish, eat street food, drink beer, and sleep". From there, you can get trailheads (Puerto de Navacerrada, Cercedilla, Manzanares, even La Granja, etc.) by train or bus in 1 or 2 hours. Train is very convenient for schedulle and frequence. And really, in Puerto de Navacerrada or Cercedilla, the routes begin in the station!
You can see some posibilities in the pages of Peñalara, La Maliciosa, Monton de Trigo and Siete Picos, here in SP. There is some routes in any of them. Also, you have some routes in La Pedriza (page in SP, too) but the trailhead is some kilometers away the village.