The easiest and quickest route to Portugal from London is via the Eurostar (www.eurostar.com) which leaves from London St Pancras station to Paris.
From Paris you take a high-speed TGV train to Irun which is on the Spanish border and then take the Sud-Express train hotel from Irun to Lisbon.
The cheapest return fare involving a couchette seat on the overnight train is €200, and whilst that might not be the cheapest method of getting to and from Lisbon, the experience will be far more enjoyable than all that airport malarkey. Overnight trains are exciting and can be very romantic, although if you can afford to upgrade to a bunk in a cabin, you will enjoy it all the more and stand more chance of sleeping.
From Lisbon you can get direct trains to the Algarve, and if you want to go to Porto, then you will need to change at Coimbra in Portugal.
Another option is to go via Madrid in Spain, which involves two overnight trains between Paris and Madrid and then between Madrid and Lisbon with a day in Madrid in between departures.
You have the option to ‘sleep’ in a seat or there are couchettes and even cabins with showers depending on your budget.
To find out about up-to-date prices, view timetables and find contact details then a good website is www.seat61.com.
If you are living outside of the UK, you an book Eurostar directly through www.raileurope.com.
If you really believe in self torture then you might consider getting to Portugal via bus. A company called Eurolines (www.eurolines.co.uk) runs services to Faro, Lisbon and Porto. The journey is a soul destroying or life affirming 35 hours or so and can cost up to £200 for a return. However, there are savings to be enjoyed if you are a student, book well enough in advance or are young enough to be considered a youth.