In addition to Lisbon airport in the center of Portugal, there's also Faro airport in the south and Porto airport in the north. Spain has many airports. To see who flies where from an airport, look at that airport's Wikipedia page.
There are also some buses between the countries and a very few train routes. As long as you realize that, door to door, you will lose much of a day between countries (unless you drive, and then you will want to start and end in the same country to avoid the drop charge), you can certainly see them both. If you don't like that, stay with one country.
For Spain, it's easiest to think in terms of regions. This is not only a geographic division - each region has a very different history, cuisine, culture, etc, and some of them have have different languages and strong separatist movements as well. The ones most first time visitors focus on are Castille (around Madrid), Catalunya (around Barcelona), and Andalucia (around Seville and Granada). Other popular ones are Basque Country (around San Sebastian and Bilbao) and Galicia (around Santiago de Compostela and A Coruña); but of course there are others.
For getting around Spain, the high speed trains (called AVE) work very well, where they exist; the network is less well developed than in some other European countries. Buses fill in gaps in the train network, and are much nicer than Greyhound in the US; don't hesitate to take them. For some long distances, particularly if there's no high speed train, a plane may work well (right now, with the high speed service to Granada still being built, flying Barcelona to Granada saves time; when the AVE is ready, this may no longer be true).
The same is true of Portugal - buses there are more useful than in some other European countries, but for going from Lisbon to Coimbra to Porto, the train works great.
