There’s a clear method, and it’s more specific than most people expect for something that ends up in the household trash.
Throw away the old dressing and other used supplies in a waterproof plastic bag. Close it tightly, then double it before putting it in the trash. For wet-to-dry dressings it’s the same: put all used supplies in the plastic bag, close it securely, then put it in a second plastic bag, close that bag securely, and put it in the trash.
Double bagged, each bag closed. That’s the whole of the disposal instruction.
The bag is also used during the change, not just at the end. Put the old dressing in a plastic bag and set it aside. Then remove your gloves into the same bag, wash your hands, and add the rest of the used material at the end before sealing it.
Laundry has its own instruction, and it’s the part most often missed. Wash any soiled laundry from the dressing change separately from other laundry, and ask your provider if you need to add bleach to the wash water. Whether bleach is needed is up to your provider, not something to decide at home.
One more that belongs here because it’s about reuse rather than disposal: use a dressing only once. Never reuse it.
Sharps, suction canisters and clinical waste collection are handled differently and vary by service. If your dressing changes involve anything beyond gauze and tape, that’s a question for the clinic supplying them.