Sara
Selecting apple rootstocks for hobbyists is usually based on the size of the mature tree. However, you can select for disease resistance and how one might perform in your environment or climate. Some are better for drier conditions, some are more hardy, some are more prone to Fire Blight, etc... If growing for apple production and making $$$ then rootstock selection is a very serious undertaking. Many rootstocks have been developed in many places - Poland, Czechoslovakia, Russia, Vineland, Quebec, Japan, and Germany. Two that are easiest for us to get in the US, especially in small quantities are Geneva and Malling along with one from Russia and one from Poland.
Rootstocks are usually marked with a letter and a number. i.e.AX - where A is an indication of who developed it and the X is a number that is the specific rootstock variety.
M-series. from the UK by East Malling research station
MM-series. from a colaboration between East Malling Research Station and the John Innes Institute at Merton
G-series.from Cornell University's agricultural experiment station based at Geneva, New York
Bud-series (Budagovsky). from the Michurin Institute near Moscow, Russia. E
P-series. from Poland
Years ago when I did my trees, the only ones I could find to buy in small quantities was the M and MM series. I wanted very small trees that I was going to train into cordons or as low horizontal espalliers. To save $$ (newly married) I just bought 4 rootstocks and 4 scions. I was going to grow those a year or two (until I could get enough scion, the buy a bunch more rootstocks. The buying of the bunch more rootstocks never happened as I was having to travel too much to deal with fruit trees. I had to give up bonsai at the time also. I selected M27 rootstock because of the very dwarfing effects. I might have selected G65 if it had been available. Google the different rootstock comparisons to select the eventual size of tree you want as well as important disease resistance. The M27 is not supposed to be resistant to Fire Blight, but even in a yard that has a pear tree that has Fire Blight in some years, the apples have not had Fire Blight. They all have leaf spot, especially since I've given up actually being able to harvest any fruit because of squirrels and have not sprayed in years.
Grafting is not hard - just follow directions. All four of my grafts took, but the next year something chewed the scion off the graft of one tree - so I am left with three. Some debate is done over when best to graft. But most say just before the flower buds on the rootstock are about to open. Active growth period is much preferred. You can cut the scion as soon as the tree goes dormant and store storing them in damp sawdust, moss or wrapped in plastic. Keep the scions over the winter in a cool damp place.
There are also some interstem/rootstock combinations that are for specific purposes, but that was far too much trouble for me. I selected the spur type apples for scion, intending to do cardons.