HatsThere are hats available to buy in the southern empty house. When you get your first achievement a mouse will move in and you will be able to buy your first hat. Hats are purely cosmetics.
Ringsthere are a few rings in the game. Magnet rings increase your pick-up radius. Golden rings increase light-radius around you. There are also rings that improve attack speed and critical chance which are buyable later in the game in the adventurer guild.
Shoesare mostly there to increase your immunity and defense. There are various ways to get them. Either you can buy them from the Adventurer's Guild or you can find a pair in the mines.