You seem to have the gist of it.
Enchants and gems are both simple things. You apply them to a piece of gear, if they can be appied to that piece, and then you never think about them again. Enchants are specific to a slot: you can’t apply a Cloak enchant to a Weapon. You can apply a gem only where the piece of gear has a socket.
Some will be cheaper than others. It’s usually worth buying at least the cheapest enchants and gems. No point in overpaying for gems for a piece you will be replacing next week. As you get better gear, do harder content, and are less likely to replace that piece of gear, you might splash out for more expensive gems and enchants.
You might also look at icy-veins.com. On each of their spec pages, they have a “Gems, Enchants, and Conbsumables” page that gives a nice simple list of what gems and enchants you should apply. That will do fine until you get to a level where you are into the expensive end of the market and carefully simming and weighing your stats.
Both gems and enchants just add stats, or sometimes enchants add effects like a chance to add an extra effect, like https://www.wowhead.com/item=200054/enchant-weapon-sophic-devotion
Use: Permanently enchants a weapon to sometimes harness Order, increasing your primary stat by 932 for 15 sec.
Look at this character. Mouse over the slots
You will see that his two rings each have an enchant and a gem added. Other items also have enchants.
Enchants are made by Enchanters, and can also be bought at the Auction House. They appear in your bags as scrolls. You right-click the scroll, the cursor changes to a hand, then click the piece of gear to which you are applying them.
Gems can be applied only to items that have sockets.
You know if your gear uses gems if you mouse over a piece and see that the description mentions a socket.
Your armoury is here:
From it, I can see that your belt has a socket for a gem.
So does the ring you got from the Forbidden Reach, but that requires special gems you get from there, and you will be replacing that within days, so don’t worry about it.
I can see that you haven’t yet applied any enchants.
So with both:
- You have this piece of gear.
- If there is an enchant for that slot, and it is worth enchanting it, look up the best enchant for your spec and apply it, taking cost into account.
- If it has a socket, look up the best gem for your spec, buy it - again taking cost into account - and apply it.
And then forget about them. 
At this stage of the expansion, I wouldn’t bother gemming or enchanting any item below 424 ilevel, since you will replace it too quickly to care. And even with 424s, I would use the cheapest enchants.
iLevel 440 is maybe the point where you would start looking at it, but we will be getting a new season in a few weeks with a big bump in available gear level, so that’s really when you want to start thinking.
Embellishments are different from gems and enchants.
While you can add replace gems and enchants on any piece of armor where they apply, embellishments occur only on armor personally crafted for you by a profession, and they are permanent.
When you order a custom piece of gear from a Leatherworker, you can specify that they should add an Embellishment. That will ofc add to the expense, or perhaps require you to provide a reagent to be used in the crafting.
If you are not buying personalized crafted gear, don’t worry about it. Here’s a guide listing the possible
If that’s not clear, please ask any more specific questions!