Furniture can be bought from either Main Street or Sludge Street.
You can buy things like wallpaper, decor, chairs etc from stores like Yukea, DIY Shop and Bizarre Bazaar.
Yukea (which is like the real-life shop, Ikea) does not usually stock the newest items. The furniture there is somewhat plain. It is owned by Moe Yukky.
Fancier items can be found in the DIY Shop on Sludge Street. The DIY Shop sells different wallpapers (like sky wallpaper), windows, doors and floors with interesting designs. It is owned by a red monster called Dewy, who prefers to be unclean.
The Bizarre Bazaar, located on Main Street, sells weird and strange furniture. Such things like, a shark in a bowl, parts of a Dinosaur, or cuddly creatures.
Better furniture should become available as you level up.