Since there is no direct train between the two cities, the best way to travel is to take a bus from Seville to Faro.
The company that runs regular buses on this route is called Eva. Eva buses leave Seville Plaza de Armas bus station in Spain and arrive at Faro Avenida da Republica in Portugal.
Adult tickets for buses from Seville to Faro cost 18.00 euros one-way and 30.60 euros round-trip. Discounts are available for children and seniors at a price of 15.30 euros each way. They can be purchased online or at the Seville bus station prior to departure.
Buses leave Seville every day at 7:30 and 16:15, and additionally at 8:00 and 13:30 from early July to early September. The scheduled journey time ranges from 2 hours and 50 minutes to 3 hours and 40 minutes. Remember that Portugal is one hour behind Spain.
There are daily Eva buses from Seville to Faro except on 25 December and 1 January, so you can travel at any time of year.