Hi there, and welcome. You'll be having a great time here.
For the materials familiar to KH, like Shards, Gems, and Crystals, you won't be having too much difficulty in gathering them. They have been left largely unchanged, if not at all.
The most major change is arguably the inclusion of Stones. These can only be obtained from miniboss-like Heartless that require specific strategies. You can refer to the Item Synthesis guide thread for more details (I'm sorry I can't post links right now as I'm writing this via mobile).
There isn't any particular point of time when the special Heartless would appear, actually. You can find them as soon as you've gained access to their spawn area, which means you can start farming for Stones early. However, it might not be wise, considering how difficult slaying them can be especially in lower levels.
Actually, the hunt for the Stones isn't particularly hard if you know the right tactics for each Heartless (eg. Strike Raid or Dumbo for Gigas Shadows, Aeroga for pot Scorpion and Jet Balloon, etc). It's just it could get tedious due to the sheer amount of Stones you need.
I think that's why most would wait until late into the game and have obtained the spells that would make it easier. Besides, some of the very high-end items need Stormy Stones, which only Neoshadows drop, which again will only spawn in the second last room of the game. I guess there's no absolutely getting rid of this late-game final desperate scramble, although it can be mitigated in a way.