Tower of Belem, The discoverer's monument, and the Castelo de Sao Jorge are all a must. You can do them all in a day with either one of the city tours or by making your own way using Lisbon's extensive transport network. Although you'll have to bus it, or take an overland train to reach Belem.
Sintra and Caiscais both provide convenient day trips to a forest paradise and a colourful fishing town, respectively. They're around an hour out of Lisbon and well worth a day trip. The views from the Royal castle in Sintra is spectacular, just make sure you take the bus up to the entrance. Despite a vague estimation from first glances, after an hour and a half of walking up-hill you begin to realise it's not "just 'round the corner."
Cascais has fish, a plethora of nice restaurants and a fantastic promenade which links the town with the resort of Esturil. Enjoy a beer on the beach and then maybe head up to the casino, if you have money to throw around, before taking the train back to central Lisbon.
If animals are your thing, make your way to the zoological gardens (jardines zoologico metro stop) or the Oceanarium in the Parques de Nacaos (Oriente metro stop). The futuristic architecture alone makes a trip to the Parques worthwhile, as the geyser-like fountains will make the most committed technophobe pause for a photo.
Quick Tips:
If you've left the guidebook at home and are pushed for time and money, then why not pick up a leaflet for one of the guided tours at the local tourist office. Lisbon is a very user-friendly city with cheap efficient public transport. You can still see all the same sights but for a fraction of a price.
GET A LISBON CARD - 22 Euros for 72 hours gives you free transport, free entrance to all the city's museums and reduced entry into many of the capital's top attarctions.
If you're on a reasonable budget and not a vegetarian -- eat out, a lot! Steak in Portugal tastes divine and is cheap, cheap, cheap. You can get a meal for 2 with a bottle of wine thrown in for under 30 euros.
Best Way To Get Around:
Don't bother with taxis, when it comes to public transportation, Lisbon is second to none. You can buy a day pass on the metro for 1,40 Euros, and the buses or trams are equally cheap. Overland trains service the rest of the city. You need to leave from Rossio to get to Sintra and Cais de sodre for Cascais. The Ferry will take you out to Cacilhas overlooking the river Tagus.
The airport is about thirty minutes outside of the centre, while several of the buses via the main terminal, 91 is the most direct. Starting at around 7:30 in the morning during the week, they leave the centre every 20 minutes or so.