Alfama is the oldest and perhaps most medieval of Lisbon's central districts. It is hilly, twisting, and has some of its original Arabic influences. Its most noticeable landmark is the Castelo de Sao Jorge, but the area as a whole is interesting to explore on foot. Perhaps the best way to start is to hop on the historic number 28 tram to go up the steep streets. You will be amazed at some of the narrow spaces that the creaky tram will squeeze and grunt through. Once you have reached the top, you can go up to the Castelo de Sao Jorge. Otherwise, you can wander downhill along the tram's serpentine path or branch off judiciously along some cobblestone alleys for some surprising explorations.
If you have the time, try to wander into and around the Castelo de Sao Jorge, which is set on a hill and can be seen from many points in the city (the Elevador de Santa Justa and the Praca da Figueira come to mind). The Visigoths built the castle in the 5th century and the Moors enlarged the area in the 9th century. The castle and nearby royal palace were strongholds for the Portuguese royalty from the 14th to 16th centuries. Some may scoff at visiting this castle, as it has been heavily restored over the years. However, you can get some very nice views of Lisbon from the top. Amaze your eyes as the walls and ramparts tumble down the hills, sort of like a Portuguese version of the Great Wall of China. There is free admission to the 10-towered castle.
As you wander down, you may encounter a miradouro or two. This is basically a level lookout perched aside the hilly terrain of the Alfama, some of which have cafes or other facilities. Two of the more notable ones are the Miradouro de Santa Luzia and Miradouro da Graca. There is a 2,000-year-old Roman theater along Rua da Saudade.
As you keep meandering down the hill, you may wander into the Se, the 12th-century Romanesque-style main cathedral of Lisbon. The heavier style of the Se (it had previously served as a fortress) is not nearly as elaborate as the frilly Manuelite stylings of the Mosteiro dos Jeronimos or Torre de Belem. You may run into many of the working-class folk who reside in this area, some of whom still fish daily as a way of life.