Planning Your Time
You'll want to give yourself a day at least exploring the bairro of Alfama, climbing up to the Castelo de São Jorge (Saint's George's castle) for an overview of the city; another in the monumental downtown area, Baixa, and in the neighboring fancy shopping district of Chiado, and perhaps also in the funkier shops of Bairro Alto. Another again could be spent in historic Belém, with its many museums and monuments. Note that many are closed Monday, and churches often close for a couple of hours at lunchtime. It’s worth nothing that national museums offer free entry Sunday until 2 pm.
There are other attractions dotted around the modern city, and families will appreciate the child-friendly attractions of the Parque das Nações, the former Expo site. All in all, there's enough to do and see to fill a week—though if you're staying that long you should think about visiting Sintra or other sights in the hinterland.