So I'd like to actually do some pickpocketing so I can get that perk that gives me +100 carry weight before I look for the Steed stone, but basically, as this is a new game I kind of don't understand what contributes to my being stealthy and improving my pickpocketing.

Any tips or tricks to make myself better at it?

For pickpocketing, you can pickpocket a NPC, take all his stuff (if you can) and put it back in. It is a bit slow, but it helps raise your pickpocket level. You could also get someone to teach you the perk you want to raise, be it stealth or pickpocketing. For stealh, get an ennemy's attention and hide (it is preferable to choose an enemy that is far and so that you can use your bow. Less chances of him finding you!) or walk around in stealth mode.

Hi!

Is there a hierarchy to what I can steal at low levels versus high? Like do I have to start out with coins first then items, and then larger items? Do they have to be asleep or awake?

Hello!
They can be asleep or awake, it doesn't matter although I think that pickpocketing an asleep NPC is easier, but I might be wrong. There is a 'hierarchy' of items easier to pickpocket: coins are usually easier to steal than...well...pretty much everything. When you're looking at someone's inventory, the objects you can steal will have a percentage under them. The percentage represent your chances of stealing the object. 0% means that you have absolutely no chances of stealing it. Don't worry, pickpocket can be hard sometimes, but it's worth it
EDIT: keys are easy to pickpocket!

Sneak attacks will grant the highest increase in the Stealth skill. One can also just crouch behind an NPC ( like a merchant ) and move against a wall to increase the skill.

Pickpocket is based on value. Using a follower and starting out with a small amount of gold, like 25, will get things going. Just reverse pickpocket the gold and then pickpocket it over and over. Increase the amount as you advance in skill.

Pending on how you feel about it, one can save and reload to ensure a continued success, until the desired skill level is reached.

Sleeping NPCs are definitely easier to pickpocket. Be sure to review the pickpocketing info on the wiki: http://www.uesp.net/wiki/Skyrim:Pickpocket

Also, keep in mind that your victims can hire thugs to take you out (it's not clear to me how this retaliation is tied to your success or failure at the actual pickpocketing). This happened to me once inside Morthal. I was waiting for the guards to help, but they never did. Had to run pretty hard and use Calm to get away.

One tip - the palaces are good places to work your pickpocketing skills. They have many residents, most of them rich enough to have good loot. If you wait until late, you can get several targets in one location. I usually enter their room, close the door, save, then try to get their loot. The Bards college may be the best location, it had something like seven targets.