Cannot figure out how to progress.

[LuSnow]

The quest "you have selected regicide" is a level 50 quest (you can check on wiki for example). However, the level 50 is more a recommendation. You can do it with lower level. For information, it's not really a quest, it's a trial. You can go for it via the Duty Finder so it's easier.

Now regarding your question, I'm surprised because I can't recall that this quest was requesting anything related to gear, even more one of a level higher than that of the quest itself. Therefore, sorry for my question, but are you sure that it's your active quest? The different duty logs can be very confusing at times. Maybe the text you see is from another quest?

Otherwise, related to your gear, there are many different ways to obtain it: market board, dugeons & other duties, quests, tomestones, scrips exchange, Grand Company (currency seals), NPC vendors, and so on...

Generally, this game is very overwhelming at start, due to the enormous volume of information to swallow. I was myself not long ago new to it. I consumed lot of times just reading stuff to understand. Take it easy. Good luck.