At first you should decide what type of traveling do you want: big cities, the country, historical towns, catholic experience, mountain, snow, coast, partying, museums... there is a very rich offer about all of this some of then depending of the season.
Madrid is a good base for traveling through Spain cause is near to its geometrical center and is really very well connected by car, bus, train and plane. Madrid have three of the world most important paint museums (El Prado, Reina Sofa and Thyssen) among other ones, the beauty of the old downtown, the marcha (partying specially at night), a true Egyptian temple in the downtown near to the royal palace and is very near to historical and architectural towns to let one day visits to Segovia, vila, Cuenca, Toledo, Aranjuez (you can choose el tren de la fresa -strawberry train-), Alcal de Henares and El Escorial. Is hard to make detailed suggestions for this places because they are full of them.
Car Rental Spain - Other places to visit
I should suggest you to visit Andalucia. Cadiz, maybe the most American of the Spanish cities (La Habana is like Cadiz but with black little people, Cadiz is like La Habana but with more `salero'). Cadiz is the more ancient city of Europe. Seville is well know by its beauty as Cordoba. Granada is a must. You should visit Granada in Spring, and I strongly suggest to read first Washington Irving's Tales of Alhambra. The rest of Andalucia is plenty of extraordinary places in villages, small towns and country. The well know by Hollywood and TV Almera desert can interest you and the near beachs in Cabo de Gata are a must if you love the coast and scuba diving. The Alpujarras, between Granada and Almera, is a completely different landscape of green and mountains and white towns.
Remember maybe the half of the country towns in Spain have their fairs in August. So it can be a great opportunity. Bigger cities as (specially) Mlaga and Almera have their fairs in August too.
Another route can be the university cities: old beautiful cities with big important in their economies which implies young cities full of activities: Santiago de Compostela, Salamanca and Granada are a must.
An important tip could be the activities related to Don Quixote. This year celebrates the 4th century of their first publishing and there is an enourmous amount of activities, specially in Castilla-La Mancha.
Like islands? Lanzarote in Canary Islands (with the a lovely weather all the year) and Formentera in Balearic Islands are said to be extremely nice. Ceuta and Melilla are two Spanish cities at the North of Africa with their own personality, mix of cultures and history. In some way this cities are islands in land.
Like the Atlantic coast? Car rental Spain recommends visiting Galicia, Asturias, Santander and Basque Country are plenty of places and country landscapes.
Like Mediterranean? From the French border of Catalua to Tarifa (where Atlantic and Mediterranean meets and create the windsurfing paradise) you'll find hundreds of coast with beachs and nice towns to visit: Barcelona, Tarragona (the old greek and roman city), Valencia, Alicante (with their main fair at june's 23th magical night of Saint John) and Almera and Mlaga. In the middle you can discover any kind of places, from the most commercial touristic to the most typical.
More places? Cceres in Extremadura is a very nice medieval city. Len and Teruel, in Castilla y Len and Aragn respectively, are other old beautiful cities.
Don't forget Lisboa is near of Madrid and that city and the South of Portugal can be a nice complement to a visit to Spain.
And the final tip: any tips list for visiting Spain is always incomplete. Have fun.
PS:
Don't matter the place you are in Spain, try the local specialties and discover the Spanish dishes. You'll win.
