Those crystals stress me out. When should I use them, who should I use them on? I'm still on the first titan and I have a handful of commons and one rare. Should so save the rare for when I'm higher level, im assuming that your level affects your chances of getting higher power blades...
I just finished Chapter 4, and I'd say to go ahead and use them whenever you want. I like to use them as soon as I get them, so that I can find blades I want and start using them and earning points to level their arts. Your level doesn't affect the power of the blade when you summon it. When you get a new one you want to use, just buy them a core chip from the latest town you've been to in order to bring their stats up to par. The only time I'd say to hold onto your cores is if you feel perfectly comfortable with the setup you have, because eventually new characters will join the party with only their default blade, and you can summon blades for them too.
(EDIT: I'm sorry, I misread -- your CHANCES of getting a higher-power blade can be affected, not the base power of the blade itself. However this is affected not directly by your level, but by your LCK and your Idea stats. Even so, I personally think it's much more worthwhile to draw early, get your party outfitted, and have fun playing through the game, rather than gimping yourself early on just for an increased chance later. You'll be banking on chance no matter what.)
As for which characters to use your cores on, it's mostly just about filling your characters out with certain classes and element types. Nia doesn't want TNK blades, since she's a healer and you don't want her to draw aggro, but other HLR and ATK blades work just fine on her. I have Rex set up with Pyra, an ATK blade, and a HLR blade, but I found it useful to have a TNK blade on the side, particularly in one section where Tora momentarily leaves the party. It's generally nice to have a variety of elemental blades between your party members so that you can do more types of Driver Combos.
There's a lot of systems at play in this game, but you shouldn't feel like you're going to drown if you don't specifically do XYZ. If you're finding that what you're doing isn't working out, experiment with your blade setup. Don't be afraid to use the common blades for the sake of having more options.