I got most of my information through Paragon Wiki (as usual), plus announcing myself as a newbie in every iTrial I ran for quite a while... but here is a brief overview to get you started.
For your alpha slot, you can use either "shard" components or "thread" components. On the crafting tab of your incarnate powers window, you can scroll down in the recipe panel to the right to see the "thread" option. All other incarnate slots, once unlocked through iXP earned doing iTrials, require thread-based salvage to craft.
Shards drop during regular (non-incarnate) play, and some of the shard-based salvage can be acquired as special rewards at the end of task forces. Threads drop during incarnate trials (and will drop in the new Dark Astoria solo incarnate zone, when it is released in i22), and thread-based salvage is granted as special rewards for completing those trials, or in some cases for earning badges within the trials. Empyrean merits and astral merits are also granted through iTrials, and can be traded in for threads or for thread-based salvage, along with some special unlocks for your account (see vendors to the right in Ouroboros). Once your alpha powers are complete, shards may also be converted into threads for use in your other incarnate abilities. All of this trading in (with the exception of the account unlocks) happens on the convert tab of your incarnate powers window.
There are 4 tiers to each incarnate power. T1 requires 3 common salvage. T2 requires one uncommon and two common. T3 requires one rare and two common. T4 requires one very rare and two common. Crafting the T4 power also requires two (any two) rare powers, so in order to get a T4 you do something like this: T1 -> T2 -> T3, T1 -> T2 -> T3, T4.
It takes 472 shards (for alpha only) or 2,480 threads to create a T4 power, not including salvage drops for completing objectives. Obviously, most of your salvage will come from drops, but I give you these numbers to illustrate how much more effective shards are for crafting alpha powers. You'll find it is very tempting to use your precious threads to get that next alpha tier. Resist temptation! 
Also, when working on your T3 and T4 alpha powers, you will find it useful to run the WST each week, as it will grant you a Rare shard-based component (Notice of the Well). Two of these, along with 32 shards, will combine to create the very rare Favor of the Well.
Note that your T3 powers for Alpha, Lore, and Destiny slots grant a level shift. This affects your combat level; with all 3 shifts, you will fight as if you are level 53 during iTrials. The alpha level shift is also applied during regular play, so getting that power to T3 should be a priority.
I think that about covers the basics. You may want to consider mapping a key to open your incarnate powers window, for convenience. Mine is /bind i toggle incarnates, as I recall.
ETA: For a longer guide with more numbers, try here: http://boards.cityofheroes.com/showthread.php?t=257011
I'm new-ish. Again.
@Kahzi. Pinnacle, mostly.
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Originally Posted by Fulmens
The plural of anecdote is not data.
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One other thing to note for the Alpha slot powers, several of the high level Task Forces in the game (Imperious, Lady Grey, Statesman/Recluse) will offer the Components you make from Shards as rewards. For instance, the T1 Spiritual Alpha can be made with components earned from the Imperious Task Force, the Statesman Task Force, and Gr'ai matter (crafted from 150 Vanguard merits, very easy to earn in a Rikti mothership raid)
http://boards.cityofheroes.com/showthread.php?t=257011
Dispari's guide is first rate 
@Mistress Rue
Yes, I RP and play Redside on Virtue - No, Rumors of its demise have been greatly exaggerated
Do you suck at COH/COV? not yet...
I will look into these. Thanks for your help.
-- "If at first you don't succeed, destroy all evidence that you tried."
Absolutely read the guides, but the short answer is that you get the components by running Incarnate Trials (aka iTrials). You run the trials by using the LFG tool. Or just ask to be invited to a team when a trial is starting. Depending on the server, you will probably want to hang out in RWZ or Pocket D, listen in on whatever global channel is popular for your server, and wait for someone to start inviting people. BAF and Lambda are easy ones to start with. Try to find someone else of the same archetype as you and follow them around doing whatever they do. Oh, and add League chat to one of your tabs, so you can follow the leader's instructions.
Avatar: "Cheeky Jack O Lantern" by dimarie
Hey, I'm not new to the game, but I'm returning after quite some time. Went VIP to check out the Incarnate abilities, and I've unlocked the Alpha Slot to start. However, I'm still fuzzy on how to get the components to unlock powers. Is there a usable guide here that I can follow? Thanks for any help.
-- "If at first you don't succeed, destroy all evidence that you tried."