Please guide us. We would like to leisurely visit the Algarve in Portugal. How many days should we plan? Which airport is the closest to this destination? Thank you!

I believe that Faro is the airport that accepts international flights. This is what we saw in five days:

https://flic.kr/s/aHsmENSCvb

Faro is the international airport, slightly outside of the town. Faro itself has train links along the coast and up to Lisbon. The local bus services also work their way along the coast. Both buses and trains give out towards the west end.

Days, it depends on what you want to see and the time of year.

Alentejo (the area to the north) is great if you want to visit up and coming vineyards and you are more likely to meet a true locals environment, unlike the Algarve which is a bit heavy on golf and northern Europeans.

This past fall we spent 8 days wandering in the Algarve, driving the whole length from west to east. We started on the stunning walking route of the Costa Vicentina west of Lagos with its secluded, expansive beaches (Praias da Bordeira, do Amado, Carrapateira) and ended in the white Moorish town of Tavira on the river, near the Spanish border.
Because we weren't seeking a golf/tennis/spa resort experience, we skipped having a base in the middle section, much of which we found quite overbuilt. We only stopped in the middle at Porches to purchase pottery.

From Michael's great photo album, I see that we visited some of the same spots.
We enjoyed the Cabo São Vicente lighthouse & museum at sunset (in the raging wind/rain), the Sagres fortress, the lively Saturday waterfront fish market in Olhão, the Silves castle, its wineries and the Monchique hills, the beautiful cliffs and grottos of the Ponta da Piedade, the historic center of Faro, the octopus capital of Santa Luzia and the pretty one-street town of Cacela Velha, east of Tavira.

We didn't fly in or out of Faro International, just picked up the Alsa bus there to take us to Sevilla.

Algarve and ???
As to where to continue after the Algarve, and depending on how much time you have:
1) from Castro Marim in the eastern Algarve you could drive up north to one of my favorite regions of Portugal, the rural Alentejo.
It's the land of endless cork and olive groves, horse and cattle farms, wine estates (quintas), whitewashed fortified hill towns facing the Spanish border (Monsaraz, Marvão, Castelo da Vide), the interesting UNESCO World Heritage city of Evora with its Roman temple, the marble town of Vila Viçosa, the border castle town of Elvas, and it has plentiful megalithic sites: dolmens, menhirs, cromlechs dotting the countryside.
See Michael's Alentejo photo album for ideas: