Our kitchen is our 'mud room' if you consider the dogs in/out of the muddy back yard all the time! We have a slightly textured ceramic tile in that room. The only time slipping has really been an issue is if you are 'dripping' wet with rubber-soled shoes. We also recently installed vinyl plank flooring in our dining room. It truly looks like wood! Clean up consists of either dusting or damp mop with plain water. Special cleaner if really dirty. BUT it is guaranteed for life against water damage. Bought at Home Depot and hubs did the install. Allure brand. They have 2 different grades w/different warranties/prices.
I would definitely go with some type of vinyl product and some cheap throw rugs to catch the worst of the dirt/mud. You're then able to toss them in the wash or the garbage after too much use/abuse.