best way to get from Madrid to Lagos Portugal

Any way you look at it, the trip is over 8 hours. If you click on this link to Rome to Rio, it'll detail your options.

The fastest is likely by flying Madrid to Faro, and then bussing from there. However, Iberian Airlines only runs that flight on a few days of the week. You could also fly to Lisbon and train/bus, but that will end up being pretty time consuming. I'd recommend a fast train from Madrid to Sevilla (2.5 hours), timed to arrive close to the departure time of one of the Alsa busses to Lagos (something like 9:30 and 16:00, from the Santa Justa station in Sevilla. I took that bus run last year as far as Faro Airport, where I picked up a rental car, and it was a comfortable ride. It's 4 to 4.5 hours, but you gain back an hour on the clock when you cross into Portugal. Check the bus schedules at Alsa.es.

To add to ottawanderer's nice summary, the only relevant train across the border is an overnight train Madrid-Lisbon.