A bit like skincare, I think upgrading your wardrobe is something you do over time. You can't do a complete overhal and change your style in one go without a huge amount of effort, cost and confidence.
So in your case, young mum, dogs etc, I'd focus on upgrading day to day for a start - so instead of leggings and a loose t-shrt, invest in a couple of pairs of good jeans that you can still wear every day. You could buy some fairly standard ones and that would look fine, or see what is a bit more fashionable - the high waisted, front pocket, wide legged style are really nice and look good on people your size.
Ditto, invest in a couple of tops/t-shirts that are just that much nicer than a bog standard t-shirt, and find ones that fit well. Maybe something with a pattern, or an interesting neckline or shoulder detail. As we're getting close to winter, one or two nice jumpers that are still good for everyday can really up your game without making you feel like you're overdressed.
Also buy a couple of pairs of shoes that aren't basic trainers - doesn't have to be fancy, I have a few pairs of ankle boots from Sainsburys that I wear constantly. They're comfortable, relatively inexpensive and I can slip them on very easily for day to day wear.
Then, as you feel a bit more confident, think about times when that sort of casual comfort is less important. Do you regualrly go out for sunday lunch? Perhaps some nice trousers that make a bit of a statement or a casual dress?