One thing I didn't properly research is exactly how skills are leveled up. If I have a skill on my hotbar but don't use it, does its skill line level up? or do I actually have used it to kill or damage an enemy for it to receive exp?

also, do skills only level up from fights, or if I get exp from other methods (completing quests for example) do the skills I have equiped get leveled up?

Weapon skills are levelled by using the weapon, same as armour skills.

Class skills you need to have one on the bar, but this will level all class skills of that line equally. Also the more class skills of that line you have on the bar the faster they level (same with equipping numbers of one type of armour).

And npc guild skills are levelled by doing the right things for the guild - for example collecting lore books (mages guild) or killing daedra/closing anchors (fighters guild).

I noticed that you do gain lvls in the skill lines by gaining xp from quests, I think that this applies to all of the skill lines(class/weapon/armour etc..).