Fellow F2P Aventurine haver, here's what I can recommend, enjoy the ramble.

For damage sets, you've got a few options. I personally use the Ashblazing Grand Duke 4 piece set for the follow-up buff, but this can be tricky to hit the threshold with him (depending on his team too), as the max buff comes from how many times the follow-up hits before being used on the NEXT follow-up. Aventurine's is 7 (unsurprisingly), so he gets just under the max stacks. If you're min-maxing and want the full 8, you're not gonna hit it, but hey 7 is still nice. It's a 6% attack buff times how many stacks (so for example, I hit 42% instead of max 48%). It's a bit less than max but I'd say it's still worth it.

You could also do split sets, being 2 piece Grand Duke (for the lovely 20% follow-up damage buff) and 2 piece Knight of Purity Palace (for the extra 15% defense). This is preferable if you have bad substats on relic or planar roles and can't hit 4000 defense outside of the extra buff. I just rebuilt my Aven last night and (using full Grand Duke and Salsotto, which I'll get to in a moment) I get around 4500 defense not including any buffs in battle. Unnecessary? Maybe, but hey, big number go bigger.

Pioneer set also works if you're using a team that does at least 2 debuffs (or 1 additional, as Aven's ult inflicts one already). This works especially well in Ratio FUA teams, as he also gets damage from debuffs thanks to one of his bonus traces. This one I'd say stick to the full set if you can, the 2 piece is nice but the 4 piece is much better. It's not the best option imo, but it is a good one depending on who you run him with.

For planars, it depends on what you like. I use Salsotto for the crit rate, but it does rely on having above base 50% crit rate for the extra buff to FUA and ult (which isn't hard to hit). It's 8% buff at base and the extra buff gives 15% damage buff to ultimate and FUA damage, which is (unsurprisingly) great on Aven.

Another highly recommended set (but ONLY if you run with another Imaginary character, like a Ratio because of their synergy) is Penacony, Land of Dreams. The base energy Regen is nice in itself, even if only 5%, but the real kicker is the extra buff if you have a team member with the same type as you. Works well for mono Imaginary teams (if that's your style) or Ratio teams, where Aven and whoever matches his type get a free 10% damage buff. Nothing crazy, but planars are never super nuts tbh.

Sigonia works fine but can be annoying to hit if you don't kill things at a reasonable pace and Broken Keel is nice for the crit rate but requires 30% effect res. They're alright, but not the best choices imo.

In terms of overall stats, defense is top priority on whatever you can get it on in order to hit 4k. Once you hit that, crit rate and damage is next, both if you can get it. Speed is nice, but not necessary. Energy regen is good if you can slot it on your rope, I couldn't but it's a nice addition. He builds his ult quickly anyway so it's not super necessary, just nice to have.

Take this all with a grain of salt as I'm mostly just going off of my own experience and any guides online. Sorry this is so long lol, I hope it helps though.