Recommendations of best area to stay in, day trips, must see, how best to get around.....
I highly recommend staying at Residencial Florescente. It's very centrally located. There is a metro station that is less than a 5 minute walk. You can also walk to most places in central Lisbon. It's located on a pedestrian street with lots of restaurants/cafes. Staff is very friendly and it is very reasonably priced.
Definitely recommend going to Belem for the day. You can take the trolley (Rick actually gives directions for getting there in his book). I suggest getting there early and avoiding the weekend, as it can be quite busy and you'll waste a lot of time standing in line.
If you are into food and wine, there is a great tour from Inside Lisbon that runs most evenings. Our tour guide was Tiago, and he was fantastic. It's a really great way to see a lot of the city and find some great places to visit again on your own.
The metro is very easy to use. You can buy the Via Viagram card and ride as much as you want within a 24-hour period. It covers the metro, trolley, buses, and funiculars. It's definitely the way to go if you plan on using public transportation.
I definitely suggest getting Rick's book. We took one day and did his recommended walking tour on our own, and had a lot of fun doing it.
Have a great time! Lisbon is one of our favorite cities!