Welcome to the forum!
I have relatively little practical experience with the DX (cropped-frame sensor) Nikons, such as yours, so I will let other members recommend specific lenses. Your camera lacks a focusing motor, so unless you are willing to focus manually, you must use AF-S lenses that have the focusing motor within the lens. The older AF and AF-D lenses, some of which are still in production, require a camera body with an in-built focusing motor, in order to have auto-focus ability.
I tend to use a Canon 135mm f/2 lens to photograph our dogs, with a "full-frame" 5D camera. If I use a wider focal length lens, the dogs become a bit camera-shy; the 135mm allows me to keep a sufficiently discreet distance. A fairly equivalent angle of view would be achieved with an 85mm lens on your DX camera body, and Nikon's AF-S 85mm 1.8G has an excellent reputation. You can test whether you like this angle of view, by using your Dad's 70-200mm at 85mm.
I may well acquire an AF-S 85mm 1.8G, myself, eventually.