- Level & distillery spirits
- Weapons, their upgrade level, & items used to upgrade weapons (the "Something X" items)
- Secret Skills
- Most vigils (see below)
- Leaderboard scores
- Momentos & associated vigil
- Buildings in the Bastion & associated vigil
- Access to Who Knows Where
- Completion of Proving Grounds & associated vigil
While you'll have all your weapons for the new game, you won't have the ability to swap right away. Because of this, if you pick up a weapon during the story, you'll be stuck with it until you find an Arsenal. This adds a little motivation to upgrade weapons you don't use often that are also part of the first part of the game, where Arsenals are scarce and don't appear in each level shortly after you find the new weapon (like they do in later levels).
You mentioned that you don't want to finish the proving grounds if they will reset. While they will reset, you still get to keep the skills, items, and fragments collected by completing them before the game resets. Depending on how worried you are about updating or getting skills, it may be worth it to you to complete at least the vigils for your go-to weapons anyway.
Bonus Fragments on Level Completion
Since I built the Shrine in my New Game Plus, I've been playing with all but two of the idols active. I've also been receiving 2500 bonus fragments when I finish a level. I think it's tied to the idols and not some combination of idols and New Game Plus, but I can't confirm since I never played with more than one idol active in my original play-through.