Hi Larelea,
I would have the best of both worlds. I would have hardwood floors put in the family room. Then you can add a large area rug for the seating area. This helps to warm up the room and to add color and style along with your furniture.
From one who has had dogs with carpeting I want to go back to hardwood or wood look tile in the busy rooms of the house including the kitchen.
You can add large rugs in the kitchen as well. At the sink area, stove and refrigerator. These are nice when standing to cook and clean. With these rugs you can switch them out. It's also better to have vinyl in the kitchen. I lived with carpet in the kitchen and it's terrible if there is a spill. Small rugs can be
changed with the seasons and your whim along with matching towels.
Pam M