We've stayed in Principe Real and Rato and walked into the city centre (around 15-20 minutes) to access the good public transport ststem when we've needed it - rail, bus, tram and Metro.
We'
Have just done four posts on my blog on our week in Lisbon (July 2015). There's a mix of tips about Lisbon plus some names of some decent mid-range eating places and some shots of Lisbon, Sintra and Cascais.
Access www.wisepacking.me and the top post will be part IV - scroll down and you'll soon find part I..
We'd recommend getting hold of Pocket Rough Guide Lisbon as a starting point as to what to see and do - beats LP IMHO
We found our accomodation on www.booking.com and booked into Lisbon Dreams Guest House in the Rato area. We've also stayed in Casa Oliver and Pensao Londres in the nearby Principe Real area.
We stayed in Tavira for a week in March 2016 - one of the quieter towns on the Algarve and a place that gets a few mentions on here from time to time. This was our second visit to Tavrira as we had a couple of nights there in 2013. Some info on wisepacking too.
Cheap flights London to Lisbon may be a big ask given the time frame.
Try www.skyscanner.net as a starting point... But be advised that I've just had a starting price of £209 per person for return flights - out on 6th July, back 16th July and that was with budget airlines and didn't include hold luggage (we fly hand luggage only as have virtually everyone we've met on our travels in Porugal.
Also note that some of the airports marked as London xxx are some distance from either Gatwick or Heathrow airports..
Enjoy - we have, which is why we've made three trips to Portugal over the last year.