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.
If you can get the arcane smithing perk you can modify unique weapons like the nightengale bow. This will allow you to do more damage with the gear they have instead of relying on enchanting your own ridiculous bows.
Alchemy is always good to dabble in because the ingredients are everywhere, and making any potion enhances the skill. Traders always restock their ingredients every 48 in game hours. This means you can fast travel to riften then markarth and every fast travel will have a restock as its a 25 in game hour journey from one to the other. However alchemy can be tedious, and there are already plenty of helpful potions that fortify skills already in the game so making them isn't completely necessary unless you want to "break" the game.
Enchanting is always good because you can fortify archery on bracers thus making your bows do considerably more damage. If you wish to take this tree you may as well go the distance and get every perk. There is an abundance of soul gems in the game, the college in winterhold constantly restocks as do any magic shop and if you can get the daedric artifact Azura's Star (an unlimited use soul gem) you won't have any problem getting the skill up. Like alchemy, enchantment can be exploited to "break" the game but this takes some work and the game is plenty fun without the exploit.
My personal recomendation is level all 3 a bit, see which one you like doing more and then focusing on that tree. Smithing can get you the most all around increase as it doesn't add elemental damage which some enemies resist. Enchanting is better if you just want to supplement your skill in general without having to go back and improve the weapon itself every time you find a new shiny bow. Also the fact you can dual enchant weapons to have multiple effects makes it the other viable choice. So it really comes down to if you want to use enhanced effects or base equipment bonuses.
Best Answer
Generally speaking, spend 80% of your perks on direct effectiveness in combat until you get up around level 25 or so.
In other words, limit perk spending on all these non-combat skills (enchanting/smithing/alchemy/speech/lockpicking/pickpocket) to 5 perks before level 25. This gives you 20 perks to your main attack/defense/healing abilities.
Perks have a much stronger effect than skill levels, so I wouldn't worry much about exact skill levels (exceptions: magic skills not high enough limit your spell choices, all skills not high enough limit your perk choices).