The 28 tram is the most photographed thing in this city and the least useful. It costs three euros fifty to board, it is full before it reaches you, and every stop on its route is walkable from another stop on its route. Skip it. The city it climbs through is free.
Saturday
Start at Miradouro da Senhora do Monte, the highest of the viewpoints and the only one that is ever empty. Come down through Graça on foot, not by the stairs the map suggests but the long way along Rua da Voz do Operário, where the tiled facades have not been restored and are better for it. By the time you reach Alfama you will have passed four more viewpoints, none of which charge admission.
Lunch is wherever has a handwritten menu and no English on it. Expect to pay eight euros for more food than you want. Afternoon belongs to the ferry: Cais do Sodré to Cacilhas, about a euro and a half, twenty minutes, and the best view of Lisbon available at any price. Most people photograph the city from a hill. The water is where it actually resolves.
Sunday
Feira da Ladra runs Tuesday and Saturday, so Sunday is for the Gulbenkian, free after two if you have a student card and worth the walk through Avenidas Novas regardless. End at Miradouro de Santa Catarina at dusk with something from the kiosk. You will have covered most of the city, spent under twenty euros, and never once queued for the tram.