There's a few avenues further. You can slot materia into the level 55 set, purchase level 70 gloves after beating the Ixali beastman quests, you can turn in items to obtain the level 70 items, or you can purchase the level 60 crafted armor and slot materia into that (FAR more expensive than the level 55, but higher stat caps). If you're going for a cheap route, I imagine you could slot materia into your level 55 accessories, as those won't be replaced, and hope that with food and possible an FC buff that you have the stats to craft the items to turn in for level 70 gear. That should net you the stats to start crafting the level 60 sets (which are better as you can add materia to them, you can't to the level 70 items) and off-hand. As for mainhand, that only progresses via turning in items. There's a level 70, level 75, and level 90 variant of each mainhand, each requiring progessively harder items to turn in.

And no, you can't trade the items that you need to turn in.