The first thing that needs to be said is this: sprouting mango seeds is strictly for fun. Unless you're growing mango outside, or in a very specialized greenhouse, you'll never actually harvest mango from a seed you sprout. The reasons are myriad: mango doesn't fruit until they are mature trees, which means they are large, and perhaps just as importantly, most mango won't grow "true" from seed, meaning that they won't reliably produce the same fruit you enjoyed.
This is because many mangoes are grown on grafted stock, meaning they are grown from trees that were grafted onto disease- and pest-resistant rootstock when they were very young. Also, most mangoes are varietals of various species, so they are a bit like mules: they are the sterile offspring of two fertile parents. But nevertheless, it's still fun to grow new things, especially something as exotic and wonderful as a mango. So put aside your thoughts of harvesting fruit and go ahead: sprout that seed.
Light: Mango seedlings require bright but not direct sunlight. Once the plant starts to grow, give it as much light as possible, including moving it outside if possible.
Water: Regular water, but do not leave them with wet feet. Mango, like many tropical fruits, thrives in periods of alternating wet and dry. Seeds require regular moisture to sprout.
Soil: A rich, peat-based potting soil with excellent drainage is beneficial.
Fertilizer: Feed with a weak liquid fertilizer throughout the growing season. Cut fertilizer back to once a month or so in the winter.
Professional growers typically graft mango, while backyard growers often air-layer their trees to get the desired tree. In an indoor home setting, you can try to sprout a mango seed from any fruit you buy at a grocery store.
To germinate the seed, very carefully remove the outer hairy husk to reveal the inner seed. "Polyembryonic" plants will have several smaller seeds inside, while other plants will have just one seed. This seed can be suspended over water, like an avocado seed, or it can be planted with the bulging side up in a pot of seedling soil. It should sprout within two weeks. Seeds need to be kept above 70˚F to sprout and need to be given plenty of water.
Mango seeds are best sprouted at the beginning of the growing season. They will grow fairly fast throughout that first growing season, but should not be repotted until the beginning of the second growing season. Mango will grow into small trees fairly quickly (about four or five years). A mango will not fruit for at least the first four years.
Mango is one of the most heavily bred plants in the world. Aside from the basic species (of which there are about 40), breeders and growers have created thousands of cultivars. Cultivars are bred for their fruit—mango trees are not ornamental. If you're growing from seed, don't expect the fruit to be true to the parent plant.
Mango will grow into a small tree with green, oblong leaves that are somewhat ordinary in appearance.
Mature plants can easily reach 60 to 100 feet, depending on the species, but indoors, you'll be lucky to keep the plant alive for a few years. They cannot tolerate freezing. Mango sometime suffers from pests, including mealybugs, aphids, and mites. Signs of infestation include tiny webs on plants, clumps of white "powdery" residue, or visible insects on the plant. Treat infestations as soon as possible to prevent them from spreading to the rest of your collection. As always, start with the least toxic treatment option first, only progressing to more serious chemicals if your initial efforts fail.