In the people I've been teaching, they refuse to pick a champion to main (one of them finally picked Kindred a few days ago. ????????), and thus I see them struggling with individual champion mechanics - They will spend most of a game trying to figure out Ryze's combos, before dropping him and picking up rengar, or whatever; instead of learning about warding/lethality/mr/mixed builds/whatever.
Basically, I would suggest picking someone (doesn't matter who, but generally mages are more versatile, in my experience), and maining them hard until you know the basics of the game. Then I would suggest following Stevobrine's plan, and branch out into a small pool of champions.
Also, it isn't necessary that you only play your 10 champions, but I think its better if you only play those 10ish in ranked. I have had several experiences with 1st times/new-at-Xchamp in ranked, and I won very few of those games.