1. Port Tasting
Do a tour of a port cellar or two. Most of the port producers have a building in Vila Nova de Gaia just across the river from Porto itself. Not sure where to start? Want the experience of a guide? See this Porto Wine and Food Tour. In addition to Port you'll also try several local dishes.
Not all port is the same and not all port tasting tours are the same either. Each tour typically includes a tasting a of a white, ruby and tawny port with the price of admission (which can be free at some cellars). The more you taste, the more you will recognize your favorite style.
Most of the cellars have a tasting areas where additional port can purchased. This is a good opportunity to explore different ports within your preferred style, from that Port producer. For instance, I am a fan of Tawny's, particularly 10+ aged tawny's.
Then visit Vinologia, a bar that specialises in learning to appreciate port and is not attached to any particular port producer. Now you hopefully know the style of Port that you like, you can compare that type across different producers. This is where you can really zero in on finding your favorite port. They have 'vertical' tastings if you still don't know the style you like and 'horizontal' tastings to help you explore the different producers of a single port style.See more here about Vinologia on Trip Advisor.
2. Take a Guided Tour
A guided tour is always a great way to learn about the history and culture of your destination. Here are couple of suggestions, the first being a group and the second giving you the flexibility to go as you wish.
See one of Porto's oldest districts then sample some of Porto's famed port wine at Vila Nova de Gaia on this Oporto by Day - Bairro de Se Tour and Wine Tasting.
If you still want the knowledge of a tour guide but want to choose what you want to see when you want to see it, try this Hop-On Hop-Off Tour of Porto.
3. Ponte Luis I Bridge
Probably one of the most iconic structures in Porto, not only does Ponte Luis lend to the landscape, it also allows for amazing views of both Porto and Vila Nova de Gaia. You will want to go across the river to Vila Nova de Gaia, whether to visit the Port cellars or just take in stunning views of Porto itself.
See here for more about the Ponte Luis I Bridge on Trip Advisor.
4. Ribeirinha Promenade
Porto, its many bridges, and its neighbor, Vila Nova de Gaia, across the river offer up some stunning views. One of the best places to see them is along the Ribeirinha, a promenade that takes you along the river Douro and past many cafes and restaurants.
Start near the Sao Francisco Church,then head towards the river. You can grab a cheap snack or drink and take in some of the views or continue towards the Ponte Luis I bridge where there are plenty of restaurants.
Feel like sampling some of Portugal's great wines while sitting on the Ribeirinha? Check out Avepod. It has a good selection wines and ports from independent, small winemakers, as well as some delicious nibbles. See here for more information about Avepod.
See more about the Ribeirinha
5. Serralves Modern Art Museum
If you like modern art, then this is a must-see. This museum has rotating exhibitions and amazing gardens for when museum fatigue sets in (don't miss the art deco villa in the garden).
The best way to get there is by local bus. Some people also combine it with a trip to Casa de Music, since you can take the same bus to both.Find more information on the Serralves Modern Art Museum site.
6. Porto Cathedral (Se)
Porto's cathedral, or Se in Portuguese, began construction in the 12th century but was not completed until 1737. As a result, the cathedral has many different types of architectural influences including Gothic, Romanesque and Baroque.
The cathedral is also located right among several other sites, including the Pelourinho statue, the Medieval Tower and an archaeological museum. Plus, you can catch some nice views of Porto nearby.
See here for more information about the Porto's Cathedral.
7. Try a Francesinha
Porto's decadent Francesinha is a sandwich typically filled with various types of meat, covered with cheese and then drowned in a tomato and beer sauce. Haut cuisine it is not and it may give you an instant heart attack, but it is tasty. Don't worry vegetarians, you can get in on the action as many places have vegetable fillings as well.
If you want to try the Francesinha and learn about other Portuguese food, try this Porto Food and Wine Tasting Tour. Learn about Portugal's culture and some history through it's food and drink.
Of course, if you aren't feeling so adventurous, you can seek out some of the lovely local seafood that Portugal is known for.
One of the best and most popular Francesinha spots is at Versa em Pedra. Best to share one, but if you are going solo, you can order a half. More information on Trip Advisor about Versa em Pedra.
8. Jardim Palacio de Cristal (Crystal Palace Gardens)
Porto's original Crystal Palace has been replaced by a sports pavilion but the real attraction here is the grounds surrounding it. As you head towards the river, you will see all different gardens. From these gardens, you get great views from different angles of the Douro River and Vila Nova de Gaia. Not overrun with tourists, this is a great way to really relax but still get some site-seeing in. It's also nicely shaded, helping you to keep cool during the summer months. Make sure to explore all the grounds, as they wrap around the river.
See more about the Palacio de Cristal on Trip Advisor
9. Porto's Tiled Churches and the Sao Francisco Church
Porto has so many beautiful churches, and almost all are free to enter!
Grab yourself a free tourist map from a tourist office, which has the churches marked on it. Luckily, the majority of Porto sights are in walking distance and many of the churches are peppered throughout most of the major tourist sites.
Even if a church is closed, the exterior can be the best part. Many churches are adorned with Azulejos, Portugual's traditional painted ceramic work. Quite often the ones found on churches are blue and white and may depict different scenes.
The most famous church is Sao Francisco. While there is an entrance fee, most say it's worth it.
10. Day Trips
While there is plenty to see and do in Porto, it is an ideal jumping off point for day trips.
The scenery of the Douro Valley, the area where Port and other wines are made, is amazing. You can follow along the Douro river, whether by boat, train or vehicle and take in the gorgeous terraces where the vineyards are. You'll see why anyone who goes to the valley can't stop praising it.
See here about a tour of the Douro Valley.
Braga has plenty of churches and a 12th-century cathedral. It is a pleasant and walkable city, where you can fill your day with sites in town. But one of the biggest attractions is the Bom Jesus do Monte, a sanctuary, which is just outside of the city but is easily reachable by a local bus in Braga. Honestly, one of my favorite attractions in Portugal. But don't be deceived by Braga's place as the third-largest city in Portugal - Braga is small and a day trip here will suffice for most people.
See here about a tour of Braga and Guimaraes. This tour includes a stop at the Bom Jesus.
See here on how to get to Braga from Porto.
Coimbra is lies between Porto and Lisbon, making it a great stop between the two cites. However, if you are planning on basing yourself in Porto, Coimbra is only about 1.5 hour drive from Porto so you can still go and enjoy the Coimbra style Fado easily!
See here for more information on how to get to Coimbra from Porto.
See also Best Day Trips from Porto for more detail and ideas.