A good first place to grab some gear (specially when you dont want to spend gil on it!) is your Grand Company - they offer gearsets for various levels!

Other than that a pretty obvious place is, ofc, the marketboard or even crafting the gear yourself.
If you dont want pay such highprices, as some people like to ask, each area has vendors where you can buy NQ-gear. Rule of thumb here: Look at the level of the FATEs and/or leve-quests in that area, the vendors will provide gear around that level.

This is also a very helpful site: http://xivdb.com/
Check the marketboard to see which is the best item you can currently equip with that character and if you dont want to buy it there, you can use this site to look up where to go to get the NQ-vendor-version.

(Also, that title is a bit missleading... this isnt a palace of the dead issue...)