You need to have your weapon completed all the way to the end of the Pyros step. After that, you can buy the elemental +1 set. And I don't believe you need to finish the weapon to upgrade it to +2. Just finish the Eureka storyline then run Baldesion Arsenal til you get enough fragments to reach +2Note that the +1 version glows, you only need the +2 if you plan to farm BA a lot.

And yeah, for the +1 you do NOT need to fully finish the relic. But you do need to do most of it and all of the Logos actions. And probably the hardest part (aside from grinding Pagos which nobody likes) is getting all of the Logos actions. There's like 54 actions I believe that you have to get from combining Logograms. I highly recommend having a tracker such as https://ffxiv-eureka.com/logograms up at all times when getting your actions to help you keep track of what you have and what you need. Also, if you have gil to burn and don't want to farm them, you can buy the logograms off the marketboard.