There are currently no direct airline routes between Rome and Faro.
Most airlines, with the exception of TAP, Lufthansa, and a couple of Scandanavian operators, are low-cost and therefore, one would have to buy two one sector tickets via a hub and hope the flights connect.
Vueling.com, (a Catalunyan low-cost ), begins seasonal flights between Barcelona and Faro from 20 Junee 2016. They also operate Rome to Barcelona. Unusually for a low-cost they allow multi-sector bookings. I have used them in the past to travel Sevilla to Napoli via their Barcelona base...all on a single ticket.
There is a flight available on 20 June as follows:
Roma (FCO): 09:55
Barcelona: 11:45
Barcelona: 13:55
Faro (FAO): 14:50 (-1)
Another option, but not lowcost, is with IAG's Iberia Group via Madrid Barajas. Iberia operates the sector Roma to Madrid Barajas and its subsidiary Air Nostrum will be operating three flights per week in the high season between Madrid Barajas and Faro.
On 1st June, 2015 when flight start there is availability on the route as follows:
07:45h - Roma Fiumicino (FCO) Terminal 3
10:20h - Madrid Adolfo Suarez Barajas (MAD) - Terminal 4
Duración: 2h 35m
11:35h - Madrid Adolfo Suarez Barajas (MAD) - Terminal 4
11:50h - Faro (-1)
Duración: 1h 15m.
Current price is around 250€ at time of checking. See: http://www.airnostrum.es/
The final option would be by TAP Air Portugal. See http://www.flytap.com/
At present there appear to be three to four connections per day via Lisbon in early June for prices around € 175.00 at the time of writing.
However, I would issue a major warning about booking with TAP so far ahead. The company is currently in the final phase of yet another attempt at privatization. The Pilot and Aircrew unions are all making mutterings about new strikes in common months as things come to a head.
