Lisbon: what can we say... One can argue that Porto is older or that Coimbra is more authentic. The fact is that Lisbon is one of the top-10 most visited cities in Europe. That says something. It's about 245 km from the Algarve to the centre of Lisbon. So if you have the time, go out early, pay the tollway A2, cross the spectacular 5km long Vasco-da-Gama-bridge, and you can have a great day in the Portuguese capital. For a one day visit, we recommend: Torre de Belem, Padrão dos Descobrimentos, Praça de Comercio, and a good walk through the streets that lie behind that.
Évora: some 150 km to the North-east lies this “open-air museum”, a Unesco World Heritage site. The city has a nice layout, a friendly atmosphere, and is full of monuments. The oldest one within the city walls is a Roman temple (“the Diana Temple”) from the 1st century; the oldest one outside the city walls is probably the “Cromeleque dos Almendres” which is estimated to be 7,000 years old and is often compared to Stonehenge (though less stunning on arrival, impressive anyway and better situated). There is no direct motorway leading to Évora and once you have travelled the scenic route, you will be happy there isn't. The views of the Alentejo region are breathtaking and you will find yourself stopping many times to take pictures. It's 200 km (IC1 northbound, follow Beija, then Evora). Alternatively, take the A2 northbound and then the A6 towards Spain (direction Evora), which will cost you about 60 km extra plus about 22 Euros toll (one way).
Seville: this beautiful Spanish city is only 235 kilometers away and very worthwhile a visit. Moorish influence everywhere, just like the sour orange trees that it is known for, the Giralda-tower, the world exhibition neighbourhood and the site of the Ibero-American exhibition where many of the pavillions are still worth a visit (Plaza de España is a must). And then there is the Isla Mágica parque, very nice, lets say that in terms of quality and setup it is somewhere in between Disney and Zoomarine (right here in the Algarve). Great for kids and grown-ups alike.
Madrid: the Spanish capital is 750 kilometers drive away, so not suitable for a one day trip. That would be a shame given the vast array of monuments and world-class museums it has to offer. But if you are staying here for many week, a visit to Madrid should be considered. The road to get there is good, hotels (book in advance through internet to get real good deals!) and food are not too expensive. Our favorites: Spanish breakfast in the typical places you find in all streets (book the hotel without breakfast, it'll be cheaper and really adds to the experience), Museo del Prado (especially the Goya-collection), Museo Reina Sofia, Museo Thyssen-Bornemisza, the Plaza Mayor, and of course, one of the daily set lunch-menus (order from the menu of the day and you might get surprised but seldom disappointed).