Market Square- This is a place to go and see the famous "Bobby" statue, but more importantly you can find delicious local food here. Outside of the actual market there are many food stands (serving vendace) and bar/restaurants.
Hupisaaret Park- A relaxing park with natural beauty everywhere you look. A great place to take a walk on the way to the Oulu Art Museum.
Harald- This is a viking themed tourist restaurant. Some locals told me that the food is good, so after seeing the signs in different cities of Finland I decided to try it in Oulu, and the food is good! I had reindeer, moose, duck hearts, and some other interesting meat items and they everything was fantastic! The service was good, and the atmosphere is fun.
Sokeri-Jussin- This is fine dining in a rustic atmosphere. You walk to an island, and the restaurant is in and old timber warehouse. The smell of wood is wonderful and the food is traditional and good. I did find it to be a little overpriced, but overall I enjoyed the experience.
Pizzeria Da Mario- I heard that Oulu is the "pizza capital" of Finland, so I decided to try it. Every now and again you just want to take it easy and bring a pizza back to the room, and that's what I did. The pizza was good (in the NY style I like, thin and a little greasy). Even better, I got to see a glimpse of the northern lights that night for the first time, so I am glad that I stayed in and ate some pretty good pizza.
Tahitornin Kahvila- A perfect café to enjoy a coffee in the observatory at a historical site where the Oulu castle was once located before it's destruction. I loved having a coffee here. The service is friendly, the coffee good, and the building is beautiful.
Valve- A cultural center with a cafe. They have unique local art here, they have a dj playing music, and they have delicious lattes (not easy to find in Finland). A welcoming place surrounded by art and music is where I feel most comfortable. I would most likely be here every day if I lived in Oulu.
Alko- A little tip for you if you don't know; Alko is the only place you can get wine or spirits in Finland. You need to pay attention to their hours, because they are not open all the time. They have a nice selection of wine and liquor from all over the world. I tried Fisk, a fish shot, from here for the first time, and I like it.
Lapland Hotel Oulu- This hotel was super comfortable, in a good location, and it had a great breakfast. Some of the rooms have a sauna, and I recommend you go for it and book a room with a sauna.