The stretch of this new bike route heading west from Faro makes a good ride for a long weekend, or even a week if you want plenty of time to stop and do other things. From Faro airport, cut north through the village of Montenegro until you find a blue line painted along the edge of the road. Follow it onto the bike route and past the university campus. The blacktop doesn't last for long and you will soon be riding on gravel trails - harder going, but at least it's traffic free apart from the occasional agricultural or commercial vehicle. After a spell in the woods you will pass lush green golf courses en route to Vilamoura, a resort town which has bars and restaurants should you be ready for lunch or a drink by now.
Continuing westwards, there are plenty of holiday resorts where you could spend the night. Out of season many hotels close but in a place the size of Albufeira or even Gale you are sure to find somewhere that is open and has rooms available in winter. Another possibility is to stay out of the towns in one of the agroturismo (rural tourism) hotels that are starting to spring up in the Algarve. The Quinta do Mel near Olhos de Agua is one of these, and it is just a short distance from the bike route. It serves only local produce in its restaurant, including herbal teas grown in its own garden, and the night-time silence is absolute. This is very much a place to chill out after the day's exertions. If you leave Faro airport in the early afternoon, you will be at Quinta do Mel before dark even if you take things slowly.
You can see here how to do this trip to the Algarve
After a night in the middle of nowhere you may feel ready for a little more excitement at the end of your second day. The commercial city of Portimao fits the bill as it has shops, restaurants, bars and a riverside location but, being a short way inland, it is not touristy. It even has a bike repair shop, in case you need any spares or maintenance. Portimao is a fairly gentle day's ride from Olhos de Agua. To get there, you follow the Ecovia away from the coast to the town of Lagoa and then across to the River Arade. You will end the day on a back road approaching the small town of Ferragudo, just across the river from Portimao. There's a fine view as you cross the bridge.