Well as far as gearing up goes, only the current relic series matters. You need to complete shadowbringers AND the Stormblood Ivalice alliance raid series (this second bit can be tedious to get queues for as people try to avoid running it in alliance roulette by deliberately lowering the gear level, squeenix needs to implement some sort of fix to curb this). Then a quest called Save the Queen appears in Kugane's market district and you go from there. Stage 2 of the weapon series is actually set to come out next tuesday (and unlike stage 1, it'll presumably be a proper grind).

The older relic weapons are purely for glamour purposes at this point. If you still wanna go for it (as a side note, for the first two relics mentioned below, I highly HIGHLY recommend being at least 10 levels higher than the weapon involved so you can unsync the content that will be required. You will likely go insane if you try to do the Zodiac weapon sync'd):
Zodiac:The ARR weapon starts in Hyrstmill in the northeast of North Shroud starting with "A Relic Reborn". Note that the Zodiac weapon is easily the biggest and most tedious grind of the relic weapons, AND will cost 1-2 million gil if you don't have crafters at level 50 with the crafting tomes unlocked (something of a grind in itself, though not nearly as bad as it used to be).

Anima: The HW relic starts in the Idylshire tomestone shop with the quest "An Unexpected Proposal". As far as time sinks go, the Anima weapon is much less intensive than the Zodiac weapon in a sense. The biggest grind about the Anima weapon is that it requires a metric butt ton of poetics spent on various materials. You'll cap your poetics like 5-10 times in the process. As for actually running content, you basically just have to run multiple dungeons and trials once and run stage 9 of Alexander Savage like 20 times (once again, be at least level 70 for this, 80 is better. You can often find farm parties for this particular step in Party finder, and if there isn't one up, you can make one yourself and people will generally join. You can probably get by with about 5 level 80s). Also you can skip the first step of the Anima weapon if you have a completed zodiac weapon by sacrificing the Zodiac weapon (this is actually ideal, because it lets you buy replicas of the Zodiac weapon's various stages from then on without needing the completed product in your inventory, normally have to have it on you to buy them), but don't complete the Zodiac weapon JUST to skip this step, it takes MUUUUUCH longer to do a zodiac weapon than to do the first step of the anima weapon.