The quest to repair the Broken Spear is offered by Vanhas, who can be found in the small observatory room on the upper level of the Refuge. Go all the way up to the upper levels, and when you reach the entrance to the Watchers' Offices (where Zaheed is), head left and go through the dark corridor. You'll find Vanhas in the small room at the very end.

The Avikan Shock Core is currently unused, but will be part of a quest for obtaining a Unique weapon later on. The Centensian Relic is used in a number of quests, including the "Restoring the Past" questline. Hold on to them for now!

I found Vanhas, but he doesn't give me the quest. He just tells me to let him know if I find any ruins or something. What do I do now?

Originally posted by Nameless Nate.j2:

I found Vanhas, but he doesn't give me the quest. He just tells me to let him know if I find any ruins or something. What do I do now?

Do you have the Broken Spear in your inventory when you talk to him? The three requirements for the quest are:
-Have the Broken Spear in your inventory
-Must have completed the Avikan Custom Tailoring quest (completes automatically after completing or skipping the intro/tutorial mission as an Avikan)
-Must have completed the Glitch mission

Ok, another item i forgot to mention is the vault key. Is that used in any current missions?

Originally posted by Nameless Nate.j2:

Ok, another item i forgot to mention is the vault key. Is that used in any current missions?

The Vault Key is a vanilla item used to open the portals to procedurally generated dungeons called Vaults. You can sometimes get a key from a cache, so it's possible that you obtained a key before unlocking access to the Vaults.

Is the Avikan Custom Tailoring mission only available to the avikan race? If so, why am i able to buy the broken spear from one of the vendors?

Originally posted by Nameless Nate.j2:

Is the Avikan Custom Tailoring mission only available to the avikan race? If so, why am i able to buy the broken spear from one of the vendors?

Each race has its own version of the Custom Tailoring quest, though they all appear under the same name in your quest log. If you can buy the Broken Spear from the Lost and Found merchant, it means you have completed the Avikan custom tailoring quest: it'll only sell it to you if you completed that quest.

To explain my current situation, I am playing as a Droden. I mistakenly threw away the starting weapon for the race (which i now know is the Broken Longsword) and at the time, I couldn't remember what it was. So I looked up how to get starting weapons back and i read something on here about a Broken Avikan Spear, so i assumed that's what i was looking for. I spawned in the spear using console, and waited until i could unlock the quest. I figured out later, though, that there was a merchant that was able to sell you broken weapons, and that's where i refound the longsword. In short, as a Droden, am I supposed to be able to obtain the Broken Spear by myself, or am I sorely mistaken?

Sorry about the wall of text btw. Hope you don't mind.

Originally posted by Nameless Nate.j2:

To explain my current situation, I am playing as a Droden. I mistakenly threw away the starting weapon for the race (which i now know is the Broken Longsword) and at the time, I couldn't remember what it was. So I looked up how to get starting weapons back and i read something on here about a Broken Avikan Spear, so i assumed that's what i was looking for. I spawned in the spear using console, and waited until i could unlock the quest. I figured out later, though, that there was a merchant that was able to sell you broken weapons, and that's where i refound the longsword. In short, as a Droden, am I supposed to be able to obtain the Broken Spear by myself, or am I sorely mistaken?

If you're playing as a Droden, the only ways of acquiring a Broken Spear (and the upgraded version of it) are to either spawn them in, or by having an Avikan character give them to you. Each race has its own exclusive Unique weapon, and besides transferring them between characters, there is no legitimate way of obtaining those weapons on other characters.

The current exclusive weapons are:
Broken Spear for the Avikan
Broken Longsword for the Droden
Broken Prod for the Trink
Broken Shieldblade for the Aegi