Something like gravel will be easiest to put down. Something like large flag stones will require more prep to install (if you want it done well). Larger stones or rocks will be easier to shovel snow off of but perhaps less easy to roll the trash can over. You might consider a mix of stone slabs (or pavers, flagstone) and rocks. Like this maybe:

Weed barrier is typically permeable. If it is not your path will be wet and muddy.

Shovel snow carefully. Or don't shovel it - it's just for a trash cart. How many snows here don't melt in a week? Maybe one every few years?

I would like to put in an inexpensive path by my house. It would be used for rolling the trash can in and out. DH and I can’t agree on what materials to use. Slate chips? Crushed gravel? Stone?

I have two concerns:
-if we put down weed barrier to put down crushed gravel, for example, will this exacerbate existing drainage issues? This would be for a low part of our property where the ground remains wet after a lot of rain. Could we perforate the barrier material, or would that not even help the drainage issue?

-will gravel or slate chips present a problem when it snows and we need to shovel? Won’t the path material end up everywhere?