I want to quickly train my enchanting skill to the max I can... except I can't seem to find enough soul gems. Sure, when you do the main quest, you'll get a gem here or there, but they occur relatively scarcely.

Does anyone know of any good methods of obtaining these gems?

asura's star and some item with soul binding
use the soul for enchant
and then recharge it again

Is there a quest in the qame that explains the soul gem system?

I found a couple of them and I have no idea what to do with them.

You need to be really careful in this game leveling up anything too quickly. You'll find that your character needs other skills too in order to make it in the world.

In terms of souls germs in any Dwemer/dwaren dungeon you should find alot as every creature needs one to function. To recharge one what I do is enchant a dagger with fiery soul trap (Fire damage+soul trap) and sneak up to them and get a x15 and instant soul gem.

I'm a level 85 enchanter so far, and what I do is grab/steal/buy any kinds of soul gems I can (empty ones). Since I'm also a blacksmith, I crafted my own Elven War-Axe and then enchanted it with the Soul Trap enchantment, but any weapon will do as long as you can enchant it and plan on using it. When I enchanted it, I chose the "Soul trap on target for 1 second" because that grants you a huge amount of charges on the weapon, and since your weapon will cast soul trap on every swing now, the killing blow will be able to trap the soul - No point in making the effect last longer. So basically, as long as I have some empty gems in my inventory, any creature I kill will be automatically soultrapped. After an hour of questing/dungeoneering, I had about 60 lesser souls, 30 common souls, etc. Absolutely incredible and no extra work required on your part. Hope it helps!

Join the mages guild and then either buy them or steal them in the college. You can easily get 100 of each type in there.

I found some soul gem veins in the game. Not sure how common they are (mine were found in some very remote place). Managed to even get one black soul gem from them