Steps
1. Save some seeds whenever you eat an apple or just buy some from the store.
2. Take your seeds and lay them out to dry until there is no moisture on the outside shell.
3. Cover your seeds with a damp paper towel and place them in the fridge. Check every so often to make sure the towel stays damp. Once the seeds have been in the fridge for about a month, the seeds should have sprouted.
4. Put the seeds in a small cup of potting soil and remember to water them every day (or else the soil gets dried out and crumbly). Now just wait for some growth!
5. Transplant the newly sprouted seedling to a larger pot and keep watering daily. If you want 'all-natural apples' do not add store bought fertilizer (you can use leaf mulch or compost).
6. Pick a location for your tree. Choose a planting site that gets full sun, is convenient and has a big enough space to grow (in other words, no rotten apples in the neighbor's yard). With pruning, expect the tree to eventually reach a height of 20 ft or so).
7. Transplant your seedling to the outdoors. Once your little sapling has gotten big enough that no one will step on it or think it is a weed, carefully transplant it without cutting off any roots. Best time of year to plant depends on location - in Zone 8 or so and warmer fall planting can work well, otherwise plant in spring, once the threat of hard frost has passed. Dig the planting hole much wider than the roots to allow them to grow easily. Water the tree in well to eliminate air pockets, then spread a mulch of hardwood chips or hay a few inches thick, in a 3 ft circle around the tree. This will help retain moisture and keep grass from growing and out-competing your young tree's roots for water and nutrients. Speaking of which, do not add any nitrogen sources (fertilizer, un-aged compost) when planting. Wait a month or two before considering adding a slow-release nitrogen source.
8. Let nature take care of the rest. After the first year, you can stop watering it, unless you live in an extremely dry area, in which case you should continue to water, at least during the dry season. The equivalent of an inch or two of water a week is ideal for the first year, make sure you give it a good soaking not just a sprinkle.
9. Be aware of deer! If you live in an area with deer, you'll need to protect your young tree. Deer absolutely love to browse the buds on apple trees, in some cases they'll even damage the trunk itself. Mesh fencing a little larger than the tree, secured to posts will work in most cases, in low-pressure areas spraying the tree with a purchased or homebrew spray may also be effective.
10. Prune as little as possible the first few years, so you don't delay fruit bearing. An apple tree wants to build a lot of growth before it decides to create fruit - it's way of reproducing - so let it grow till it starts to bear.
11. Wait patiently for your tree to bear fruit. Eventually (hopefully!) the tree will bear fruit. Apples are normally propagated by grafting because they do not reproduce true from seed, so your seedling tree is an experiment! It may produce fruit that tastes wonderful, or not, but either way the fruit is unlikely to resemble the apple the seed came from.
12. Care for your apple tree. Growing fruit will take pruning, thinning, and keeping insects and disease at bay, among other things. There are many good books and resources to guide you. A great place to start is your state school extension, they may have helpful summary bulletins. Remember that your apple is what's called a seedling, standard, or full-size tree; follow the recommendations for this type. Good apples can be a challenge to grow, depending on your area, so read up on the pests and diseases in your locale.