When Riku leaves the islands, he stays in Hollow Bastion. There, he has his own bedroom, courtesy of the neighborhood witch Maleficent. In Reverse/Rebirth, we can catch a glimpse of what was in this room. It's confirmed to contain a wooden desk, chair, and a shelf, which stores numerous books. Whether Riku actually reads them is a mystery. Now it's not visible, but I think we can all safely assume that there actually is some type of bed in the room.
For some time in Kingdom Hearts II, he stays with Ansem the Wise in the Basement of Twilight Town's Old Mansion. However, since he has completely immersed himself in darkness by that point, he can warp in and out as he pleases. Therefore, we generally have no idea where he stays throughout that game.