Fukuoka City is one of Japan's five largest cities and is the biggest on the island of Kyushu. It's the capital of Fukuoka prefecture, a state that is at the top of the island and spans across it from ocean to ocean. The city itself is considered great to settle down in by many, and has been ranked in the world's top 10 in Quality of Life by Monocle. As someone who lived in the prefecture (in Yukuhashi) and frequented the city often, I can't say I disagree.
The central city itself is easy to get around on the two subway lines. There are three main areas: Hakata - the central station and downtown, Nakasu - the more "adult" nightlife area with yatai (outdoor food stands), and Tenjin - the shopping and nightlife area. They're all on the same subway line and are either next to each other or a couple stations away. You could walk to each area if you'd really like to, but I wouldn't recommend it to most.
Tenjin is my personal favorite part of tone due to the abundance of shopping, food and it being really walkable. Department stores and malls are centered right around Tenjin station, but if you walk a few blocks west, you'll hit an area called Daimyo. This area has a lot of smaller shops, boutiques and places to eat.
One of the best views for photography is the Fukuoka Tower area near Momochi, a 15-20 minute walk from the Nishijin subway station.
The most famous shrine nearby is Dazaifu Tenmangu. You can get there by train from Tenjin station, and the road leading up from Dazaifu station to the shrine area is lined with shops selling Japanese confections and gifts. If you go further in from the shrine, you can check out the Kyushu National Museum, which is worth adding onto your trip as it has various special exhibits and a big glass wall that reflects the surrounding nature.
Fukuoka also has a variety of interesting festivals within the city as well as a short trip outside of it. The most well known one is the Hakata Dontaku, which is a historic festival that takes place on the streets in central Fukuoka city.
Another is the Hakushu Festival [J] which is a beautiful riverboat festival in Yanagawa city that honors Hakushu, a great local poet.
A third notable festival is the Daizenji Tamataregu no Oniyo Festival [J], a fire festival that is focused on exorcising evil spirits.
Fukuoka's famous foods are some of the country's favorites. Hakata tonkotsu ramen, for one. For food places to check out, see my Five Places to Eat in Fukuoka list.