In the case of your example, it's either/or.
You need to move up the perk trees in order, but you don't need to clear every branch.
You also don't need to max multi-rank perks to move past them. One point is sufficient to move on.
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
Armor items have a base armor rating (based on the type of armor, eg Steel) that cannot be changed - but you can improve the effective armor rating (i.e. the number that appears on the item when you have it equipped) in a few ways:
Increasing the level of the armor skill relating to that type of armor (eg increased Heavy Armor skill will improve the rating you receive from heavy armor)
Unlocking armor perks in the respective skill tree (the first perk in the tree, i.e. Juggernaut for Heavy Armor or Agile Defender for Light Armor)
Unlocking the unison perk in the respective skill tree (Well Fitted for heavy armor, Custom Fit for light armor) and wearing four pieces of the same armor type to make it work
Improving the item via Smithing - the higher your Smithing skill, the better improved your armor will be, and having a Smithing perk associated to that armor type will also help
There's also a perk you can unlock through a quest which will give you a permanent +25% bonus if wearing Dwarven Armor.
Any of these methods will take effect immediately for armor you're already wearing - you won't need to create a new set to see the benefits, although you may need to unequip and re-equip your armor items before the benefits will be visible.
Bearing all of the above in mind, there is a hard cap of 80% physical damage reduction (567 in full armor without a shield, 542 in full armor with a shield, and 667 if unarmored), beyond which you'll see no additional benefits. The cap is easily reached before maxing out all of the above methods - so, for example, it's possible to achieve the armor cap with Steel armor if you've maxed out all other benefits; you don't necessarily need Dragon Armor to do it.
Source (UESP)