For human clothing I have a duffel bag from some brand called Newhey that I got on Amazon years ago that’s still going strong, just a huge plain black bag with pockets for almost anything. I don’t think it’d fit tall boots but you could probably fit paddock boots and half chaps (helmets I just carried by the chinstrap or put on as soon as I’m out of the car at the barn but you could fit one in there if there’s not a ton of other stuff) in the shoe compartment, I’ve fit ankle boots that are basically the same proportions as paddock boots in there before.You could probably fit a few (for sure at least a flat curry and a hoof pick) brushes in the side zipper pocket on the outside - I can cram a hair dryer in that pocket on mine, it’s a squeeze but it fits.
As far as a specialized bag for horse gear I know they make some backpack style bags for again, the human stuff, not so sure about the grooming supplies.
I had a little wooden crate that I put a really dark colored wood stain on for my grooming stuff but I just wanted it to look fancy and while I did take it back and forth to the barn back when I still had a horse it was more to keep my stuff from getting mixed up with everyone else’s stuff (crummy boarding barn, lot of communal brushes and mine looked similar) and wouldn’t be as practical as a bag. (Looked really pretty though, now just sits in my closet with the brushes in it for whenever I may need it again…)