Good quality cat tree furniture is probably as important for your cat as its bed or feeding bowls. It gives your cat somewhere to climb, rest, stretch and sharpen its claws on. It can also help save your furniture and curtains from being scratched.
Cats need to scratch at something to help them keep their claws sharp. They need them sharp so they can catch prey such as mice, to latch onto a tree when they are climbing or leaping or as defence or aggression if they are in a fight with another cat.
I strongly suggest you invest in a sturdy scratching post when you first bring your kitten or cat home. If you are able to afford it, make sure it has ledges of varying heights and sizes so your cat can lie up there and watch what's happening or sleep on it. Cats love being up high, they feel safe from small children or dogs and they enjoy watching us mere mortals. If you can't afford everything at once, make sure the cat furniture can be added to at a later stage. That way you can keep adding to it as you can afford it.
Don't worry that the cat furniture looks so huge compared to your kitten. Kittens grow incredibly fast and it will 'grow into' the furniture quickly.
Any cat furniture must be strong enough so your cat can stretch out and sharpen its claws without it toppling over. And it must not become unbalanced when your cat leaps onto the ledges. If your cat gets frightened when using it or feels insecure, you will have wasted your money. I made this mistake when I purchased a scratching post for my cats. The first time they used it, the post fell over and they have refused to use it since.
The variety of cat tree furniture available is fantastic. I suggest having the scratching post section covered in something durable such as sisal. This is ideal for sharpening your cat's claws. It may start to look untidy after a while but if your cat is used to this texture, it won't be interested in your carpet as that will 'feel wrong'. If you don't like the frayed appearance, it is a simple matter to cut the ends off. Make sure any ledges are large enough for an adult cat to comfortably curl up on. If you have more than one cat, make sure there are multiple ledges or, if you can afford it, one cat tree for each cat. If your cats are happy to play together and share, one should be sufficient.
If you have your scratching post or similar from when your cat is still in the kitten stage, it will be much easier to train him to use it. If you already have an adult cat, you can still train it to use the scratching post but it will take longer. I suggest that each time you see your cat about to sharpen its claws on the carpet or your chair, pick it up and take it to the scratching post or cat tree furniture. You can also rub some catnip into the material on the post to attract your cat to it. It will take time and patience, but cats are very smart and will learn. Try hanging toys from the various levels. These can be simple feathers on a spring or even just some scrunched up paper or a ball of old wool tied to one of the levels.
Once your cat is used to having its own cat tree furniture to play and sleep on, it will become a well used and loved piece of your cat's daily life. And the rest of your furniture, carpet and curtains should be safe from those claws.