Shortish trip to Spain

I have made six trips to Spain in recent years, and have seen many places, thanks, in part, to friends in Asturias, who love to play tour guide when I am there. We have been known to do the Salamanca, Cordoba, Granada, Sevilla, Toledo loop in six days, but it wasn't ideal (just a "sampler" trip). I'm guessing you are flying into Madrid? Have you seen anything there before? I find something new to do there every time I go. Since you lived in Salamanca, you know the best culture and people watching is hanging out in the local plaza mayor into the evening, wherever you are. The only time I was in Galicia was to go to Playa de los Catedrales as part of a day trip with my friends. Keep driving distances in mind, and remember if you are going north, winding through mountains is going to take more time than driving on flat land. I'd be happy to help out if I can give you any pointers.

Yes, love Salamanca. Also, Avila, Toledo and Segovia if you haven't do those places. Santiago de Compostela is worth a visit.
Other great places in Spain are in the north, the Basque region as well as Rioja (wine country).

In 7 nights in Spain, you can do Madrid - Salamanca - Santiago de Compostela and back to Madrid for your flight. 2 to 3 nights in Madrid, 1 or 2 in Salamanca, 2 or 3 in Santiago, final 1 night in Madrid.
My personal allocation would be 3-1-2-1 nights but you could do 2-2-2-1 or 2-1-3-1. 3 nights in Santiago lets you do a day trip somewhere else in Galicia.

A car is not needed at all. Madrid to Salamanca is quick and easy by train. For Salamanca to Santiago, you can take a frequent bus to Zamora and the train from Zamora to Santiago. Train again from Santiago back to Madrid.