In our entryway (which is not a mudroom, but I wish it was) we have shaker hooks at two levels - upper for the grownup stuff, lower for the kid stuff, but high enough for snowpants. Hooks are assigned per kid. Snowpants get hung under coats. Outdoor items without fabric hanging loops or arm straps get a hole cut in the tag by for the purpose of easy hanging! (Or you could sew in a loop.) IME, things that are hard to hang don't get hung, or they fall off easily.
I also have separate hooks for backpacks/bags.
We have a small bedside-size dresser with four drawers. Each kid has a drawer. The drawer is for dry hats and mittens and scarves and balaclavas. (BTW, if your kids don't have woolen balaclavas, I highly recommend them. Ruskovilla is the brand we have and they are marvelous at keeping us warmer.)
Also, I do the spot system - kid #1's clothes have one Sharpie dot on the tag, kid #2's clothes have two dots, etc. Makes it fast and easy to sort out whose is whose and where they should go (even little ones can do it, though we had some confusion when my #3 was 4yo and my #4 was 3yo because they kept thinking spots = age and not birth order...but I digress.) Even hats and mittens get spots. Dark items that won't take sharpie, or knit things, can be spotted with black or white fabric paint (squeeze-bottle).
We have a two-level long low shelf for shoes and boots...boots line up on top. We also have a large overflow "shoe/boot" basket for extra stuff like rain boots (not used often in winter, but sometimes in a thaw, which comes unpredictably) and extra shoes.
My next project is to install one of those drying racks with all the arms that fold out perpendicular to the wall and fold back down. For drying mittens/hats/scarves/socks on right in the entryway.
Also, we have a large rug in front of our door. People with wet clothes stay on the rug until their wet stuff is off. So they don't track wet slush everywhere.
And we have a wooden drying rack for wet outerwear. Unfortunately it's in another room, but it's still easy to remind them "wet stuff goes on the wooden rack." When everything is dry, it goes back on its hook or in its drawer, and then when it's time for us to go out again, it's not so complicated.
Also, leave plenty of time for getting ready to go somewhere... Also if I have them, I keep extra snowpants and mittens/hats in the car in case someone gets terribly wet and needs a change, or a mitten gets lost, or we are in a rush and forget something.