Instructions
Play through the game until you reach level 8. Go to Costolot's General Store to buy a cactus seed. You can also talk to Seedus, the laughing man who wanders around the garden. He offers random seeds, so it will take some time getting a cactus seed using that method. You can also use the seed pouch to purchase the cactus seed.
Go back to your garden and plant the cactus seed in either soil or sand. Both surfaces are available in your garden.
Wait for the seed to grow into a cactus. You can use three green fertilizers on the cactus seed immediately after you plant it to accelerate the process.