I'm biased because I have an apartment there, but I think Alvor is great. Huge sandy beach with a number of little coves on the Eastern end, lots of very nice and very reasonably priced restaurants and very family friendly.
There are a number of private transfer firms that offer a trip from the airport to Faro for around €50 for up to four people. The journey takes around 50 minutes. Or you can save money by going for a shared transfer, but this can take a lot longer because of drop offs along the way.
You'll get lots of information on Alvor at the following link