Here's what I'd do:
There's so much potential in that closet so I suggest clearing that out first and sorting like items with like and tossing whatever is no longer in use. Giving the inside a fresh coat of white paint and adding new shelf coverings such as oil cloth or adhesive vinyl will freshen up and make it more likeable.
Here are the items that need storing and my suggestion of where they go.
Recycling - At the bottom of the closet I'd set up a recycling station for paper (11) and bottles in their crate. Hoping the mop bucket (12) will fit down there too. Other recycling can go in tubs (8) on one of the shelves above.
Pot lids - My suggestion is to get a standing rack (5) for them on one of the shelves instead.
Plant dirt, gecko & frog dirt, grilling coal - Get a couple of nice looking containers with lids on (4) that can be out in the open by the entrance and place the bags in there.
Gardening supplies, cleaning products, paint - Go in tubs and storage containers/boxes ( 6+8+9). Like things together. The ones with handles make it easy to take a kit of something with you to another part of the house. Use a label maker (3) to mark each container at the front so everyone knows what goes where.
Plastic bags - If there's room on the inside of the door I'd attach a plastic holder (2) there but they could also go in a tub (8) on a shelf.
Frying pans, vases and jars - Go on one of the shelves again grouped by type.
I'd also consider adding a simple wine rack (10) in there somewhere so those bottles are out of the way too.
I think this is basically just a matter of using the space to it's full potential and making things more easily accessible. Let us know how it turns out, Maureen!
1) Broom and brush holder - My Amazon Store
2) Plastic bag holder - IKEA
3) Label maker - My Amazon Store
4) Bin with lid - IKEA
5) Pot lid holder - IKEA
6) Container with handles - IKEA
7) Mop and broom - IKEA
8) Clear plastic tub - IKEA
9) Wooden box - IKEA
10) Wine rack - IKEA
11) Newspaper bin - My Amazon Store
12) Bucket - IKEA