I would recommend Lagos any day over Faro,
Faro, although the capital of the Algarve, is really an administrative centre and not a travel destination. The old centre around the Sé (Cathedral) is small.
Lagos, however retains the feel of a historic Portuguese town while having great beaches as well as restaurants and night life. The west coast, if you have a car, is well worth exploring.
I must declare an interest here....I live in Lagos.
As regards hostel accommodation, the Pensão MarAzul is great but be aware that a dirty great big car park is being built right behind it at present.
If you are hostelling and on a budget have a look at the following.
http://3monkeys.me.uk/monkeyhouse/
(Good food at night in their cafe next door )
http://www.risingcockhostel.com/
(In-crowd destination - But you are always well looked after by Mama)
http://juventude.gov.pt/Portal/Lazer/pt/PLagos.htm
State-Run Youth Hostel
http://www.goldcoasthostel.com/
Gold Coast Hostel
For information on getting to Lagos, have a look at the articles on the following page on my blog:
http://barlaventostravelblog.blogspot.com/search?q=Lisbon+to+Lagos
