Biking around the city of Nakatsu is an absolute delight. The entire city center is accessible by bike, and if you are adventurous, you could also cover some of the outskirts areas by bike.
To make it even better, you could rent a bike for free! And conveniently, you could simply grab a bike from the train station, and explore the city of Nakatsu right away.
To rent a bike, you need to go to the information center of the station, which is just near the main exit. Ask for "rent-a-bike" and the person at the information counter will understand you, despite of his English proficiency skills. However, I was told that they do have English speaking personnels at times. The person will give you a form to fill, and you need to present him/her with an identification proof. After this, she/he will take you to the bike parking lot, show you your bike, and will give you the key to the bike lock. When you are done using the bike, you have to park the bike at the place you originally took it, and then return the key to information desk. And that is as simple as it can get.
You now have a bike to explore the city for however long you want. Not only is it very safe, but it's also very comfortable to cruise around with your bike. Nakatsu has bike paths at almost all areas, especially the city center. Being a bike friendly city, you won't find the need to use any other public transportation in Nakatsu. If you want to visit Rankanji temple, the scenery around the river, or other outskirt areas, you need to rent a car or take the bus.
As you can see from this short video, I personally had a great time biking. I made it to Nakatsu castle in 8 minutes, and Somo shrine in 20 minutes. You could also try biking on the path that overlooks the sea.
The biking experience is definitely one of the things that will bring me back to the remote, but beautiful city of Nakatsu. Nothing can beat the experience of gliding along a path that overlooks the sea, a castle, 9 temples, some shrines, ancient japanese style houses, the green rice paddy, and much more.