The relic quest does give a job related title so there's that. Besides that, yes, just use drops, quest rewards, and vendor items (in that order/priority) on your way to 60.
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There are weathered weapons, but not weathered versions of relic weapons.
For relics, there's an unfinished version you get during one of the earlier steps of the initial quest and its appearance looks just like the completed relic, just lower stats. You can buy another copy of your relic weapon from the calamity salvager if you've already earned it in-game before. Or if you've gotten through the entire upgrade chain through zeta, then you can buy replicas of the zodiac weapons. But you can't buy a relic (or replica of it) if you haven't earned it.
I believe Heavensward MQ will give better gear.