Apple trees have very interesting genetics. I read an article on them years ago in Scientific American, and no two apple trees are EVER genetically identical. That is the reason we have so many different varieties of apples, and they never breed "true" to the parent tree. Only by grafting apples can you ensure consistency. All the common varieties that we have come to know, are grafts. Boring, I know.
That does NOT mean you can not get a good apple from any old seed. You WILL get an apple tree, just not the one you thought you would. Could be better. Could be worse. Apples are in the same family as roses. Interesting. But think of the rose hip left over from roses in the fall - it looks just like a tiny apple. Cut an apple across and see the star in the apple - a five-pointed structure. Very nice. Good to show kids.
Apple seedlings that sprout spontaneously on your land from a discarded core, obviously love your land, or they would not be growing there. This MAY show good root stock, and may indicate a good specimen to graft on to. If you want to wing it and see what kind of apples you will get, be my guest. But, if you already have a favorite tree, think about using that root stock specimen you found and graft on to it.
Remember - if you will keep the root stock tree in place and allow it to grow where it is after grafting, there is no need to root prune. BUT if you intend to dig up the spontaneous sprouted tree, you may want to "root prune" it for a few years prior to moving it, to keep the roots from going every which way.
For good future transplanting: Plan ahead. Try to keep the roots in a clump. Do that by taking a shovel and pushing it down VERTICALLY into the earth - say - 18 inches to 24 inches from the trunk - every so often for a few years. When it is time to dig up the tree to replant it, remove the dirt beyond the circle of vertical shovel (slices? whatever they are). Excavate from the OUTSIDE, leaving the ball intact and one heck of a big hole in the ground around it.
Moving a big ball of dirt is no problem - just do not think you are going to carry it. It can be dug up, and rolled carefully and placed on a strong piece of canvas or thick, slippery plastic and dragged to your site as if on a sled. Hopefully you have selected a perfect place to replant your tree, and have dug an equally huge hole to accommodate it. When you drag your tree and lower it into the ground, REMOVE THE CANVAS/PLASTIC and place some nice rich soil around it. Do NOT stomp on the soil to compact it - YET.
This is where the advice of an old farmer comes in handy: Despite what it may look like, run a hose directly INTO the newly added dirt where your tree is and where you have put in some soil around it. It will turn into a mud pit around the ball - that is GOOD. Do not disturb the ball itself, just water-in the dirt. Stuff the hose into the mud/dirt, round and round. See the bubbles? That is what you want to get rid of. You do not want hollows under your tree. The water will then wash dirt close to the root ball. Add dirt as needed. Stick the hose into the dirt again, making sure the tree is nicely vertical and settled. There is almost no need to stomp around the tree ball. No breaking roots that way. Just add as much dirt as the tree wants and water it all in. Repeat as needed. Keep the dirt at the same level as it was around the tree, where the tree was originally planted.
This works well for all planted/transplanted bushes and trees.