This is an occasional bug that many people (myself included) have experienced. For reasons currently unknown, a skill will be debuffed down to its starting level for your character's race. Increasing your skill level (whether through normal skill usage, training, reading a skill book, etc.) generally seems to restore the skill to its proper value.
If you're on the PC version, you can also fix it by opening the console and issuing the command player.modav skillname 1 followed by player.modav skillname -1. Note that the internal name of the Archery skill is "Marksman", not "Archery".
As you noticed, Destruction is pretty straightforward. Each cast and while continued casting, you will increase skill, but this has a caveat: only if you are attacking someone. If you're fizzling your spell in the middle of town, it's not going to increase it.
Alchemy is also straightforward: each potion created will increase your skill. As dpatchery notes, eating ingredients for effect determination will also increase your skill by a small amount.
With Enchanting, each enchant (or disenchant, as LessPop_MoreFizz notes) as will increase skill. dpatchery also notes that you can increase your skill by recharging items with soul gems.
Restoration, namely the healing spells, only increase skill if your health is below its maximum. The others only work if you're in combat or near hostile enemies: this includes spells like Steadfast Ward.
For summoning spells (essentially most things in the Conjuration school), you won't get a skill up for the summon until you enter the range of hostile mobs and, in the case of summoned creatures (like the Atronach and the Familiar), they do some damage. With bound weapons, entering range of hostile mobs is all that's necessary to gain Conjuration skill; however, using them in combat will increase their appropriate weapon skill, not Conjuration.
Alteration and Illusion spells that are targeted towards messing with hostile mobs need hostile mobs within range to skill up. Buffs, like Oakflesh, don't increase skill without hostility as well. Others, like Candlelight and Muffle, are recastable without hostility but only increase skill by a small amount.
Best Answer
Everything that affects bows affect crossbows as well (damage multiplier etc), even
Deadly Aim. Those perks that aren't immediately obvious work like this:Hunter's Discipline- more bolts (from mobs who drop bolts in the first place)Ranger- move faster while aiming and reloadingQuick Draw- faster reloadUpdate: The mechanic of the weapons can be confusing.
Quick draw yields faster draw for bow and faster reload for crossbow. Ranger lets you move faster while aiming with your bow or crossbow. The thing with a crossbow is that you aim an already loaded weapon and with the bow the draw action is part of the aiming. In the end, it's no real difference.