How exactly am I leveling up?[both user level, and skill level]

Do they both use the same criteria? No?

I could have sworn that my skill raises with consecutive skill use. But then I also notice my skills go up randomly when receiving a quest reward.

And individual powers and abilities. How exactly are they leveled?

Apparently any skill you have equiped in your skill bar will level up that skill and also that skill tree. So when you get experience you get experience for everything in your skill bar, I think.

Also any weapon you use will skill up even if you do not have that weapons skill on your bar.

I was wondering why some of my stuff was leveling even though I never used it more than a few times.

Same experience. I just tried to explore as much as possible, which is fun when you are level three and wild craturers at levels ten or eleven are chasing you. I got to level five just by exploring and even my heavy armor and magic increased even without using it. I found that odd, the leveling is fine, but your skill should not level up that easily.