Pretty much everything carries over apart from iterms that progress the game. You might as well spend your insight, but I’m fairly sure the shop stock carries over to NG+
Chalice dungeon progress should stay, and they aren’t affected by the NG+ difficulty bump like the main game is. So you can come back later on NG+ if you like.
It’s worth hoovering up any late game weapons/ tools, like the Chikage, Rakuyo, Bowblade etc. in case you fancy trying them out for NG+. You could check a guide to see there’s not any optional bosses you’ve missed.