The most common way to get into Fanning Island is by taking a boat from Christmas Island (Kiritimati). It's also quite common (although not as much in recent years) to visit the atoll as a port of call from a cruise line. Holland America has it as a humanitarian stop on their way to French Polynesia.
There is a dirt road that completely circles the lagoon. There are small dirt trails running in every direction from the road, but be mindful of the native population's personal property and space when hiking the trails. There is a modest collection of bicycles near the lagoon's main dock and you should be able to rent one for a small fee.
There is a beautiful beach to swim in that has an area with soft sand. Other areas have large coral pieces that can be difficult to walk on. There are also areas that farm seaweed.
There is a concrete basketball court on the island that was built by a cruise line years ago. The men of the island enjoy a pick-up game and are friendly when approached about playing. Be mindful, however, that the men do not stop smoking their hand-rolled cigarettes, even while actively playing the game.
You will not find any stores on the island in the conventional sense. There are natives that are willing to sell you certain trinkets though. A popular item with visitors is a woman's "defense knife." It is made from palm wood and is studded with shark teeth and other decorations scavenged from the sea. American currency is generally accepted.