Quick answer
What is the best way to follow the Douro River in Porto?
The easiest way is to start in Ribeira, cross or admire Luís I Bridge, walk along the Gaia riverside, enjoy the views back to Porto and, if you have more time, continue towards the western river areas and Foz.
For a slower experience, combine the Porto side and the Gaia side. Porto gives you the old city feeling. Gaia gives you the postcard view back to Porto, especially around the riverfront, Jardim do Morro and the Port wine cellar area.