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. I break down Fukuoka city into three main areas: Hakata - the central station and downtown, Nakasu - the more "adult" nightlife area with yatai (the famous outdoor food stands the city is known for), 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 town due to the abundance of shopping, food and it just being really walkable. Department stores like Parco and Daimaru 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. The feel of it is a bit like the side streets of Tokyo's Harajuku/Omotesando area, if you've been to Tokyo before. At night, there are bars and clubs, as well as some yatai food stalls in case you don't wish to venture next door to the Nakasu area.
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 relatively well known festival, Hakata Dontaku, that takes place on the streets in central Fukuoka city. You'll see people dressed up in traditional wear carrying and pulling big wooden structures, shouting chants and spraying water.