Forget about wool, silk, or delicate material rugs - that's a straight disaster with three pets in the house. Also avoid really light colors because they're going to look stained way sooner than you think. What actually works well is going for something with tough synthetic fiber like polypropylene or nylon; these materials are almost impossible to stain because they don't absorb liquids the way wool does. Plus, the hair stays on the surface instead of getting embedded, so cleaning is way easier with a vacuum.
My recommendation is to look for rugs specifically designed for pets - they exist and they're a game changer. Pick dark colors with patterns or textures that hide stains (dark gray, brown, beige with patterns work amazingly). Hair's still going to show up, of course, but the point is the rug itself doesn't look destroyed. Also, grab a cheaper rug the first time; if you see it holds up well, then you invest in something pricier later. Some even come treated to be stain-resistant and odor-resistant, which helps a ton with accidents.
Don't forget about maintenance: vacuuming two or three times a week is key, and keep a pet-specific cleaner on hand to tackle stains right away. With that and picking the right material, you're going to be able to enjoy your living room without stressing out all the time.