Have a look at the items you buy regularly for your pet. Are they biodegradable? Do they contain palm oil?Could you switch to a less plastic-heavy option?
If you feed your pet from convenient wet-food pouches, could you feed them tinned food? If so then you'll need a lid - like this - to seal the tin to avoid smells and to keep the contents fresh.
If your pet won't / can't change from food pouches then save up the pouches in a big airtight box (first cut them open, rinse and allow to dry) and take them to your local Terracycle collection point, locations are listed here. The pouches are then washed, chopped up and turned into outdoor plastic furniture.
Dog poo bags are often to be found in nature unfortunately, either through deliberate laziness or because they have blown away on a gust of wind when a diligent dog walker was preparing to clean up after their pet. A solution is to use biodegradable bags, like these from 'Planet friendly Poo Bags Direct' sold by Amazon, which will rot down rather than remain as micro-particles of plastic for hundreds of years.
Denzel's vegan dog treats come in compostable packaging (not sure if that means home compostable or industrially 'food waste bin' compostable yet, but I'm finding this out).
Pet ownership comes with great responsibility - even more after reading this blog post - and great joy.