While there are exceptions, basically, the rule is: the worse apples are for eating, the better they are for cidermaking and vice versa.
The reason is, the apples that are the best for eating (honeycrisp, gala, granny) are largely comprised of fructose, and some malic acid. Fructose ferments out completely.
What largely makes for excellent cider apples is the presence of tannins...which you don't really want when you bite into an apple, but you do want when you take a sip of hard cider. Otherwise, IME, you have a one dimensional product with faint notes of apple and booze. The tannins are unfermentable, stay through to the finished beverage, and really give a cider (or a wine) structure. Thats why so many people treat (or as wine purists say, "manipulate") finished wine/cider with tannins and acids.
If you are looking for specific varieties, try Rome, Paula Red, Winesap, Spartans, Wolf Rivers, Red Astricans, Northern Spies, and maybe Jonathans as a start.
The other thing you can do is use a fraction of juice from crabapples (which are full of malic acid and tannins) with juice from apples otherwise largely intended for eating. Or as said above, you can treat finished cider with some of these acids and/or tannins, which are available for purchase.
I think doing it through Freecycle is an awesome idea. You may also be able to find and/or incorporate some pear juice, which I personally think rocks, especially when highly carbonated.
All that being said, I've also made a quick cider with single varietal golden russet juice, which are good to eat, make great n/a juice, and a good fermented cider as well. Cidermaking is a lot closer to winemaking than brewing, where it is more about the agriculture than the process.
As the kids say, YMMV!