Here's the first bit of info:
Once you've finished chewing the core of your delicious apple, extract the seeds and leave to dry in a cool place until it the exterior is devoid of any moisture.
Then, place each one into a seedling tray and bury in some well-draining potting mix to about 10mm (1/2 in) deep. Cover or place the tray into a cold frame and leave until the seeds begin to germinate and sprout.
Once the seedlings have two or more leaf levels they can be potted out individually into bigger containers where they can continue to grow until they're ready to go in the ground.
Keep in mind that it usually takes 2-3 years for a nursery bought apple tree to produce fruit. So, plan to not see any results for at least 6+ years from an apple tree that's been propagated by seed.
Wonderful, right? Here's info from another source:
1. Pick an apple that you like and save the seeds. Place them in a damp paper towel and then fold it into quarters. Place the towel into a plastic bag into the refrigerator. After at least 6 weeks they should start to sprout. Only about 30% of the seeds will germinate so plant enough!We actually bought 2 winesap apple trees the other day, but since we LOVE apples, and they're good for livestock, we really want a whole lot more than just 2 apple trees. I placed an ad on Craigslist the other day for heirloom apples to buy and am getting some bites... we'll get some of the heirloom apples, eat them, save the seeds, and plant!!
2. Now plant the sprouts into small containers filled with potting soil. Keep the soil moist and in a sunny window.
3. When the plant is large enough, plant it outside when it can get about 12 hours of sunlight per day. It will need good ventilation and full sunshine. Don't worry about the cold weather as apple trees need the cold to produce fruit.
4. Take good care of the tree while it is young. Fertilize and prune at the appropriate times. You local garden center should have plenty of information.