Piece of advice, if you see something similar, don't hang around. Rodents carry lots of diseases in their poop, and several of them get in the dust and kill you that way. Don't breathe that if you can help it.
So how do you get rid of them, then?
Well first, deny them food. We figure this one is stealing dog food, so we're feeding the dog earlier, and feeding him a little less so he doesn't leave any in his bowl. This won't do much for awhile, as rodents are inclined to hoard food and it'll be some time before that runs out. It will make a reinfestation much less likely, and make step two more effective as they'll have to forage more.
Part of denying them food is also plugging whatever hole they're using to get in and out. If you find a hole or gap big enough for rodents to squeeze through (smaller than you'd think, maybe a quarter to a half inch) squeeze in some caulk and stuff in coarse steel wool. Rodents can't chew through steel wool. The caulk just makes the steel wool hard for them to push out.
Step two is get traps. Lots of traps, more really is better in this instance. They make fancy electrical traps, but they're expensive. I can get a lot of snap traps for what one of those cost. Live traps are stupid. You know those rats they teach to run mazes? Rats are smart enough to avoid the trap once they've been caught in it, and that doesn't mean they've moved out. Live traps are also stupid because rats are MEAN. You've heard the expression about fighting "like a cornered rat"? There's a reason for that. Rats are also smart enough to learn from the mistakes of others, so get lots of snap traps and move them around once they've caught something.
When you place the traps, keep in mind that rodents don't like to walk across open spaces. They'll walk next to walls, beside fences, under overhangs, or such. I stick the bait side closest to the wall, and I use peanut butter for bait. Don't forget to wear gloves when you empty the traps.
I don't use poisons. I have kids, animals and food growing in my yard. I'm not putting poison in there. Yes, I know my kids could get hurt by a trap, that's why I'm careful where I place them. On the other hand they could get dead by rat poison, so I don't have any.
After you haven't caught anything for a week, it's probably safe to assume your pests are either dead or moved out. Open the doors and windows and get a breeze going. You want ventilation for the cleanup.
Now the CDC says to do it different, and this is what I should probably have done:
What I did was drench everything in bleach water and sweep it up.
Final thought, if you've got rodents watch for snakes. One often follows the other. While you're doing your cleanup, don't stick your hand anywhere you can't see.