Each piece of equipment can be improved in 7 ways:
Fortifying
Infusion
Awaken
Insignias
Talismans
Gem Slots
Star words
I'll try to break down each step.
Fortifying gear.
The materials to forty gear can be obtained in several ways, from hunting cheif mobs for shards to combine in to the crystal to buying packs of them from the Cash shop,
How ever you get them it easy makes the difference between an average character and a decent character,
in addition to improving single parts, the average of your entire set up can bring you in an addition bonus once the average reaches +4.
Done at johnny in Atlantis
Infusing gear.
During general play you will pick up random parts for other classes that are either not worth selling or you have no interest in, these parts can be deconstructed into a currency called Ether, Ether can then be used to improve the base stats of individual equipped piece of gear,
the interface is in the bottom of your inventory under "Deconstruct" with a secondary tab in the pop up box for infusing.
Awakening the soul of your gear,
During the instance you will pick up dusts of awakening,
Each piece of gear can be improved 5 times using Dust's and gain an additional stat specific to the part.
Done at Johnny in Atlantis
Adding insignia's
Soul insignia charms can be obtained from the platinum noble daily reward boxes or purchased from many vendors in town, the insignia will add an additional blue stat to the bottom of each piece of gear.
This can be done at johnny in Atlantis
Adding Talisman's
As part of the Territory battle, factions who own lands are given leadership crystal's, these are the currency used to buy talismans.
Talismans add an additional purple stat to your equipment, ranging from stun time reduction to additional elemental attacks,
Gem slots
All gear comes with 1 open gem slot, fortifying the gear will help unlock additional slots,
At +4 you have 2 slot,
+7 gives 3 slots
+10 is 4 slots
and the coveted +13 brings 5 slots.
Star words
For gear above level 60 you can add star words to them,
each part has several different words that can be added so it comes down to choice, the most basic word for each part can be purchased from dyana in Atlantis,
Once a star word is added it has to be activated,
To activate each word there is a set combo of dusts listed.
To add these dusts, slots have to be opened,
there is a short quest that teaches you about this but many people just skip the text then ignore this option, loosing out on thousands of extra HP and other fun bonuses.
The dusts them selves have to be purchase from Arm's Crafting players, or made yourself.