Yes, you can grow apple trees from seeds.  However, most apples available today are hybrids and will not come true from seed and therefore may not resemble the parent fruit.  Crabapples are said to usually come true from seed, like the Sargent crabapple tree although there may be size variations in the fruit.  It can take ten years or more for trees to fruit from seeds.  

If you are patient enough to still be considering apple trees from seeds you can start them indoors in a foam coffee cup with potting soil.  They should be planted in a shallow hole and given plenty of sun.  Keep watering to keep them moist.  Once they've sprouted you can plant them outside.  Plant them where they will have lots of sunshine and keep them watered.  Young trees should be pruned to a modified leader system and then pruning should continue to remove diseased or dead wood and to trim the tree to the desired shape.  Apply one pound of a 10-10-10 fertilizer for each year of growth.  Apply half of the fertilizer in January and the other half in June.  The fertilizer should be broadcast evenly beneath the tree.

Your patience can be rewarded with apples that you lovingly planted and cared for.  You did it yourself!  Your neighbors will be envious!