Downtown (Baixa) is buzzing atm and it's good for shopping. That is nice for walking and there is a large pedestrian street. It has a small Interactive style museum about the history of the city.
It leads to a large avenue with top and expensive brands. From there you can get on a two section lift that eventually takes you to the castle (it's not fantastic as castles go, but it does have a nice view), or you can take a cable car.
There is a tour service called Hippotrip, with an amphibian bus, that goes around the city and in the river. There are also antique trams for tourists, and now tuk-tuks. You can also rent motorised vehicles to travel around, but taxis are fairly cheap, and the underground system (metro) for most of it is not bad. You have mobile phone signal and even free-wifi in some stations.
You can take a bus or the tube to the East (Oriente station), which has a fantastic Oceanarium (Aquarium type), and a science museum (but mostly for children), a nice garden and a large shopping centre (Vasco da Gama). You can ride a cable car along the river and there is a tower with a rotating restaurant at the top (I have been told).
To the west, you have the most historical part, Belém. Tourists usually take the 15 tram (modern, usually) from Baixa, but be careful with your wallets.
There you have a large monastery with a derived gothic style (Manuelino), which is highly decorative, a navy museum, an archeology museum, and a military tower in the same style as the monastery. There is also a large Cultural Centre with a free exhibit of modern art, from a large private collection (Berardo)
There, you must go to a small shop that sells custard tarts (Pastéis de Belém). You can buy packs to go, or walk in, along the almost maze to find a table and relax.
You can, walk to the electricity museum (free) at an old coal based power station by the river.
If you can, take a day to go to the nearby town of Sintra.