No-one cares about the school run, in the sense of it's not something one dresses for specifically.
But for the majority, it's the first time out of the house in the morning. And a bit like Flylady talking about the importance of getting dressed right down to your shoes, this is about how you are facing the world for the day.
I think, OP, a good place to start is with a good haircut that you can style easily yourself.
Think about the day ahead, and what practical aspects you need for your clothes. Comfy shoes and boots are likely to be on the list - try TK Maxx
You'll need a coat, come autumn, but they're beginning to appear in the shops now so start looking. Think about the features you need as well as the look - waterproof? (almost certainly) hood? (good idea unless you know you have a good hat or spare hand for an umbrella) bum covering? There always used to be an annual autumn thread about which are the best coats, but if there is one, it'll be in S&B
Decide which shape trouser/jeans you like best early on - it'll make a difference to what tops and footwear you choose.
Which are the bits of your body you like best?
Is part of your perking up, toning up? Because combining morning errands (whether school run or anything else) with exercise can be a good way to fit it in. If so, you'll need a sports bra, good trainers, running tights or yoga pants. Sports Direct is very cheap, Decathlon nearly as cheap, with better trainers (and fewer bad headlines about employment conditions)