You can farm Creation tomes for easy i330 gear; just do roulettes on level 70 jobs for Creation tomes. After that, you should be able to queue into either Sigmascape NM raids or Swallow’s Compass. Sigmascape drops tokens for i350 gear; Swallow’s Compass drops i345 gear.

You can also just farm Creation gear, and then do Rabanastre a few times for the Rabanastre Coin to upgrade them to i340. There is no weekly limit on the coin, so you can do it enough times to boost yourself from i330 to i335. Rabanastre also drops i330 gear that you can use if you want to buy some with Creation, but not grind as much.
I wanted to have the above quoted for what I'm about to contribute to this discussion because it is the most relevant thing to talk about. I've played MMOs before and have a decent understanding of how this sort of thing works.

Speaking as a brand new player, I didn't see it coming as well. In doing everything up to here I did hit the dungeon where I needed minimum ilvl of 300. I wasn't close to that and since I hadn't had to spend any gil on really anything, I used almost all of my liquid assets to hit that so I could progress. Now I have another big gap without any warning.

Having played other MMOs I get there'll come a point where you have to grind. it's kinda the thing you work for in leveling up and progressing. This has simply been done very poorly here. Almost universally though this happens after you have attained max level and are beginning to do endgame raids. Coming to XIV as a brand new player and having already realized the stuff I've been blowing through was endgame content at some point doesn't make the situation any better at all. It's a hard stop in progression because... well I don't know. I don't think anyone else really has the correct answer either. Probably not even the developers?

The biggest point I'm going to make is that I found this thread via Google. The reason I quoted the above is because I had NO IDEA that those things actually existed whatsoever. Without any clue that those dungeons exist, of course I wouldn't have the required ilvl to progress. Now it makes sense. It should make sense naturally through the flow of the game. Not through searching on Google and finding the poster that I quoted being genuinely helpful albeit through the rest of the thread somehow defending this. Imagine you're reading a fiction novel/book and you have to stop almost at the end to Google how to finish the book. What kind of experience is that for the reader?

I'll stop complaining here and go figure out where these dungeons are, how to unlock them and how to progress. I have absolutely no idea how to do any of that. I do appreciate the help and guidance that has been given here and none of this was meant to make anyone here feel badly. Except for the people making the game who thinks this is ok. It's not. This is jarring and disjoints the whole experience. It could and should have been done better.