1. When you reach max level, you should grind to get to the appropriate item level to do current content.
There's a lot of way to get gear now like grinding for Tomes of Scripture, doing Alexander 9-12, doing Sophia EX, getting your Palace of the Dead 235 and 255 weapons, doing your relic quest, even doing the Weeping City of Mhach for 230 gear or buying crafted 250 gear.

To do the end game stuff, you'll need an ilvl of:
-At least 230 for the normal Alexander, the Creator (floors 9-12)
-For the Savage versions of Alexander you'll need at least 245 for 9, 250 for 10, 255 for 11 and 12.
-235 for Sophia EX
-210 for the current EX roulette dungeons
-205 for Weeping City
-For Palace of the Dead, you can unlock it as soon as you have a class/job at level 17.

2. To unlock the HW relic quest, you'll have to do the quest An Unexpected Proposal that you get from Rowena in Idyllshire. You do not need to have any of 2.0's Relic quests done or unlocked to start this one up, I believe.
If you have your WHM's Zeta weapon, you can trade that in to skip the first grind of the quest for any job's relic.