Since Porto and the Algarve are in opposite directions, you do not have enough time to do justice to both and to Lisbon.
Don suggests that you spend the first three days in Lisbon with day trips to Sintra and Cascais. (Day trip trains from Lisbon run often.)
Then spend the next three days either in Porto or the Algarve (suggest Porto with a day trip to the Minho). Trains run regularly between Lisbon and Porto and Lisbon and Faro. Day trips from Porto can be by train or bus depending on your destination.
Our lodging experiences are documented in our Portugal travelogues. Ziners may have other suggestions since there are many options in all price ranges.
My thought was different from Don's in that you should consider taking a bus to Nazare (under two hours) to enjoy a small seaside experience while you adjust to the time change. From Nazare you could head up to Porto (the morning bus takes three and half hours with no changes required) for a couple of days and save Lisbon for the end of your trip.
Buses in Portugal: http://www.rede-expressos.pt/index_uk.htm Some rail information here: http://www.portugal-insite.pt/canais/listagem/canal.asp?canal=32&ling=2 Regards, Don and Linda http://www.thetravelzine.com