Here is a few picks from the Algarve´s wide array of beaches when basing yourself from Faro. This is not an exhaustive list by any means but hopefully will give you some ideas of what beaches to visit when in Faro.

See also:

• Best Algarve Beaches
• Faro City Guide
• How to Get from City to City in Portugal

The Closest - Faro Beach (Praia de Faro)

Faro has it´s own beach that you can swim at or lounge on the sandy beach enjoying drinks or snacks from the surrounding stands and restaurants.
The beach can be reached easily on the city buses 14 or 16 (which departs across the street from the main bus station).

Bus fare is around 2 euros one way and the journey takes about 25 minutes from the city center.

Praia de Faro isn´t the only beach in town. For some spectacular views and less people, you can catch a ferry from Faro´s city center (right next to the old city) to Ilha de Barreta.

Easy Day Trip - Tavira

Most of the Algarve coast is can be reach in less than 2 hours from Faro, but it only takes about 40 minutes to reach Tavira by train from Faro. The town itself is quite interesting, with some nice churches, and old town and plenty of history. It´s a good idea to try and explore the town after your trip to the beach.
See here on how to get from Faro to Tavira by train, bus and car, as well as how to get from Faro's airport to Tavira.

The Ilha de Tavira (Tavira Island) is where you will find Tavira´s beaches. Once in Tavira, to get to the island during high season there is a direct boat that takes about 15 minutes and costs about 1€ each way to the island.
During the rest of the year, you´ll want to catch a ferry from from Quatro Aquas.

The ferry journey takes about 5 minutes and costs about 1.50€ round-trip. There is a bus from Tavira´s city center that you can catch to get to Quatro Aquas. Always verify transport, particularly during low season.

Overnight (Or Longer) - Lagos

Lagos can be done as a day trip (see how to get from Faro to Lagos) but you may want to consider it as another base during you Algarve vacation. It has great beaches (along with an active nightlife) but also is a popular location to explore the eastern portion of the coast.

Here are some highlights of Lagos´s beaches:

Praia da Batata or Lagos´s "Town Beach" is the closest beach to the town center.

Meia Praia (Meia beach) is one of the longest beaches in the Algarve, making it always easy to find a spot to sunbath. This beach is a bit less commercial and has some beautiful views.

Praia da Dona Ana is around a 15-20 minute walk from city center and considered to be one of the more stunning beaches in Lagos.