Train travel Lisbon-Lagos

If you're booking 5 or more days in advance to get a discounted ticket, then only www.cp.pt can sell those. Tickets are also easy enough to buy in the station. Although they are reserved trains, I would not be too concerned about them selling out on a typical day, but there could be some risk following a weekend or holiday.

Comboios de Portugal (CP) is the Portuguese National Railway Company: https://www.cp.pt/passageiros/en/
They operate all the trains in Portugal.

Rail Europe IS NOT a train system, it is an agent in North America which re-sells tickets. The tickets they sell are for CP trains.
You choice should be purely on price and ease of website use.

For lots of info on train travel in Portugal, see here: https://www.seat61.com/international-trains/trains-from-Lisbon.htm

Definitely use comboios de Portugal. The site is pretty easy and you can get your ticket on your phone. Book your tickets from 5-29 days ahead and get 40% off. They will ask to see your passport on the train.
You do have to change trains but it is very easy. You get off one train and wait on the same platform for the next train to come.

there are also good quality and direct express buses which are just as fast to Lagos (because the train route goes way off course).

http://eva-bus.com/index.php?lang=uk

timetable and prices summary http://eva-bus.com/gen_pdf.php?SERV=1&ORIGEM=Lisboa&DESTINO=Lagos&VAR=1

Lisbon terminal is at Sete Rios near the zoo.

there is another company but their website is so poor they don't deserve business!

Thank you for all the help and advice.