Porcelain tile.

Tiles and a washable indoor/outdoor rug large enough to accommodate shoes and boots , a couple shoe trays will also be of great help .

Tile. I'm in Canada too and suffer the messy weather. Tile is low maintenance and lasts pretty much forever. Grout is not something that needs constant sealing.

Also, put a mat in the garage by the door to the house so people can wipe their feet. That will keep most of the mess out.

Vinyl sheet flooring. No seams, no grout. Dirt colored with a pattern. A long washable carpet runner will pick up and hold a lot of the dirt. (Buy 2 and you always have a clean one. ) I agree with boot trays.

Tile is good too but grout gets dirty. If you choose tile, get a dark grout.

I have linoleum (Marmoleum) in a bedroom and I'm not happy with it. It is soft and shows scratches and dull scuff marks even though it has been lightly used. My mother's 10 year old sheet vinyl has held up much better, even though it's in a heavily used kitchen and hallway.

i had to look up some sheet vinyl to see if anything is different looking from the old stuff.

I guess you can get stuff that looks like this

or vinyl tiles from Lowes

I'd look for the large (14x14") hex porcelain tiles and do small grout lines. something like this looks good and is easy to care for. use the Mapei FlexColorCQ stain proof grout.

(this might even be a vinyl tile. hard to tell from the photo. but a porcelain tile would look the same)

these are a 12x24 matte porcelain that looks like cement (tilebar.com)

these are also porcelain. something like this gives you a little camouflage too