What is the best way to propagate an apple tree? I'm thinking the best way might be to cut off a small twig and splice into another tree, but what kind of tree would be good for this?
You can splice your twig right onto another established apple tree, cutting it back enough to give the twig room to grow. Then you will have two types of apples on that one tree - this is one way they make multi-variety trees in the retail trade and is probably the easiest way to get started with your new acquisition. You can also buy apple "rootstocks" (small apple trees grown not for their own fruit, but as a base for grafting other varieties onto) from some fruit tree nurseries online or there might be a place near you that sells them. The kind of apple rootstock you purchase will determine the height and vigor of the new tree - some make smaller trees than others (dwarfing rootstocks.)
If decide to propagate this tree by grafting, then be sure to graft several twigs, not just one, because sometimes they don't "take." If there is a county extension service near you, you might ask if they have a handout you can use to help you decide what type of graft to use and how best to care for your new tree (or new apple branch) post-grafting.