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.
To add immersion to the game, Bethesda added a bunch of "world encounters" that do nothing other than to provide some flavor as you travel around.
Among a ton of other things, Imperials escorting a Stormcloak around the map is one of these random world encounters. You can ignore them or help the Stormcloak: it doesn't matter.
There's another chance to help a prisoner outside of finding a chance encounter on the world map:
When you first enter Solitude, there's a prisoner being set up for an execution. You're technically not expected to save him: it's just a flavor event to give some background to what Solitude is (an Imperial stronghold). But it is technically possible. You don't get anything other than a "thanks, bro!" for doing it.
Best Answer
The choice between Stormcloaks and the Imperials is supposed to provide a little more immersion in the story: you choose a side, and while both sides have comparable quests, they take on a different flavor and will have different actors.
To this end, both the Imperials and the Stormcloaks have and 11-part quest chain concerning the Civil War that ravages Skyrim. The first few quests are almost identical, but then they branch off as each side wins (or loses) major fronts in the war.
And in general, choosing one side or the other makes things easier for you if the side you chose is in power.
For example, if your side controls a major city, you generally don't have to do any special favors for the Jarl in order to get permission to buy a house: it'll be given as a gift for siding the cause.
But the Elder Scrolls series is all about choice, and you're not going to be limited much by what side you choose. Choosing the side you most identify with will generally work out just fine.
But fear not, you didn't make a permanent choice when you went with Ralof (the rebel). You'll be able to play both sides until the last possible moment. You don't seal the deal until you swear an oath to either side by going to their stronghold city.
Additionally, there's one more chance to switch sides during the civil war: