The best way to explore Porto's city centre / downtown is to walk. Most must see places are near each other, and while walking you will come across plenty of locals and discover some hidden streets where local customs can be observed. Porto is not a plain city, so wear comfortable shoes.
Catch a train from Sta Apolonia Station in Lisbon, takes only about 2.5-3 hrs and very inexpensive.
walking is best...but if you need to ride, bus and taxi fares are also inexpensive. Inset, a pose with my friend, Anne, in the centro area.