One day: time to see a little more
Now you can also walk across the Dom Luis I bridge to visit the Mosteiro da Serra do Pilar, see more of central Porto and have time to pop into a museum, such as the Museu Nacional Soares dos Reis, or a church or two.
You might also have time for some shopping in the Rua de Santa Catarina.
Two or three days: time to see much more
Now you can really get to know Porto. You shouldn't miss the Museu de Arte Contemporânea and Serraves park, and go to the nearby Boavista area to see the villas of wealthy and the Casa da Música.
You should also have time to visit the Museu do Carro Eléctrico (Tram Museum), and the nearby Palácio de Cristal park, the Museu Romântico and maybe the Museu do Vinho do Porto.
Also visit the Palácio da Bolsa and the neighbouring Casa do Infante.
If you can, also try to visit the Gaia district and its port lodges.
Longer: enjoying Porto to the full
If you're in Porto for longer, you can really relax, enjoy a few long lunches, see a bit of the newer parts of the city, visit the Foz area and the muth of the Douro river, and perhaps take a cruise or boat trip up the Douro Valley or visit attractive towns (around an hour away) such as Guimarães and Braga.