There are a couple of ways, and here are a few of them.  Your Trees: Every day you log in, there will be two trees on your island that will drop furniture when shaken. It can be a hassle running around and shaking 20-30 trees, so I chopped all but two of my trees down.  Mystery Islands: While you are searching for villages using your Nook Miles Tickets, do not forget to shake the trees there. On each mystery island, there will be a tree that will drop a furniture item when you shake it.  Balloons: Balloons will spawn on your island every five minutes or so. Use your slingshot to burst these balloons, and sometimes, they’ll drop crafting recipes for furniture and furniture items. The content of the balloon about the color has not yet been solidified, but any balloon is worth popping and getting the reward. Nook’s Cranny: Make sure to drop by Nook’s Cranny every day. Timmy and Tommy will have two to five furniture items for sale every day. The chances of finding something you don’t have every couple of days are pretty high.  Wisp: Now and then, you can find a ghost wandering the island at night named Wisp. When you speak to Wisp, Wisp will lose five spirits because you scared him. Once you find all five spirits and return them to him, he will offer you a reward. You get two options, something new or something expensive. For the sake of getting more furniture, I would suggest something new all the time.  Bottles: Once to twice a day, you can patrol the shores of your island in search of washed-up bottles. When you find these bottles, they contain crafting recipes. But if it ends up being a recipe you already know, they sell for a couple of hundred bells.  Your Villagers: Now and then, when you speak to a villager, they will gift you a piece of furniture. They can even send you furniture items through the mail. Another special event that happens between your villagers and you are games. Usually having something to do with cards, if you win their games, they gift you with furniture or clothing items.  Other People: While I was in my ANCH mood, I joined four different Facebook groups. Each of the groups had times/sections where players could chitchat about turnip prices, island visitors, and what type of furniture is for sale in Nook's Cranny. I had decided I wanted to make a diner on my island, and I wanted to use the pieces from the diner set. I had only had the table and bar, but chitchatting with other players helped me gain access to the entire set in a couple of days.