Lisbon seems to live in a happy time warp, the 21st century in a Belle Époque setting of tree-lined promenades and pushcarts selling hot chestnuts. Creaky old wood-paneled trams warn Volvos off their tracks with persistent little bells, a Lisbon sound unchanged since the late 19th century.

Those restored trams and their modern counterparts are the visitor’s best friends, connecting the oldest and newest highlights with big windows revealing places that tempt stepping off to explore. Buy a day pass (about $5) to hop these and hill-climbing funiculars at will.

For a fast start, take a cab to Castelo de São Jorge, the hilltop fortress that forms Lisbon’s backdrop. From its lofty setting, look across the city and get your bearings. You’ll see Lisbon’s hills, bounded by the wide Tagus estuary that made Portugal rich during the golden age of exploration. First occupied by Phoenicians, the castle’s walls and square towers enclose the old Santa Cruz neighborhood, worth wandering.

Leaving the castle’s main gate, follow the street ahead into Largo do Contador, where Comidas de Santiago is a good coffee stop. Below, Largo de Santa Luzia, a blue-tiled belvedere, overlooks Alfama rooftops to the river. Catch Tram 28 downhill to the Se, Lisbon’s medieval cathedral, where St. Anthony was baptized in 1195. Walk down to Largo da Madalena to shop for fine Portuguese handicrafts at Santos Ofícios Artesanato, and hop back on Tram 28 until the streets flatten out. Watch for a monumental arch ending the street to the left and get off on Rua Augusta, where you’ll find the new MUDE design museum. Beyond the arch, in elegantly arcaded Praça do Comércio, atmospheric old Café Martinho de Arcada serves impeccably fresh seafood for lunch.

From Comércio, Tram 15 parallels the river to Belém, where a striking monument recalls Portugal’s great explorers. Highlights among its riverside parks are the magnificent UNESCO-acclaimed Jerónimos Monastery (don’t miss its exquisitely carved cloister) and the resplendent royal carriages in the Coach Museum. Opposite the Maritime Museum, the new Berardo Museum of Modern and Contemporary Art shows works by Warhol, Picasso and Dalí.

Tram 15 returns, ending a few steps from Lisbon’s heart, the Rossio (once the Roman hippodrome). Stop for a glass of Dão wine at the Art Nouveau Café Nicola, where spies eavesdropped during World War II. End your day climbing aboard the Gloria Funicular, alighting at Miradouro de São Pedro de Alcântara for views of the castle turning golden in the sunset. Any narrow side street leads from the belvedere into a street of restaurants: Alafia and Senhor Bife are good dinner choices.