Faro makes a good base for the eastern Algarve. Tavira, half an hour east by train, is a quieter and prettier town than Faro, with Moorish-era streets, a Roman bridge and the wild beaches of Ilha de Tavira just offshore. Olhão is closer (a 10-minute train ride) and worth it for the twin-domed fish market and the ferries to Culatra and Armona.
Inland, the village of Estoi combines the pink rococo palace with the Roman ruins of Milreu — a half-day trip, easily done by bus. For something bigger, the Moorish hilltop town of Silves is about 90 minutes by train via Tunes; its red sandstone castle is the most complete in the Algarve. Sagres at the southwestern tip is too far for a comfortable day trip without a car but rewarding if you have one.