Two- or three-tone patterning / shading / stippling is good. Totally plain - any colour - is a nightmare.
80/20 wool / something just a little harder wearing is practical. But check density - some 80/20s seem very keenly priced, and have less weight of wool per square metre

Coconut matting at entrance is magic. Get a good big one though, that all 4 paws have to walk on before they hit wool

Hate, loathe and detest Scotchguard. IME, once dirt does show - and with pets, and a smallholding, dirt
will show! - once it does show, if it's Scotchguarded you really can't get the dirty patch clean. I think the treatment stops the washing water/soap getting into the fibres

. So my experience is that after the first year or so, carpets and furniture treated with Scotchguard look and are more dirty that ones that were not treated.
I also loathe beige carpets (in my house) with a passion. Simply not practical.
Of all the colours I've had, pale but not too pale green seemed to cope with dirt pretty well. Anything tending into orange/red/brown will too, but may not suit your décor so well.
The other thing is to look at what's
outside the entrances. If you can get some pea shingle/gravel down, that does a stonking job on cleaning and drying feet before they ever get inside the house - and that's furry feet as well as human feet, in fact perhaps even more so.