No trains connect these two areas, but both Alsa bus lines (Spanish) and Eva bus lines (Portuguese) run busses between Lagos and Seville. Alsa is faster, with fewer stops along the way, including both Faro and Faro Airport. Alsa runs two busses per day, leaving Lagos at 8:30 and 14:00. It's a 4 hour ride to the first Seville station, Plaza de Armas on the western edge of the centre, another half hour to the main train station. If leaving from Faro, it's a two hour shorter trip. Note that there is a one hour time zone change crossing into Spain, so a four hour trip gets you there five hours later on the clock. See alsa.es or eva-bus.com for full info (both sites have an English option).