A longer lens will give you more reach, therefore you dont have be be as close to the animal your shooting. Fill the frame (like you have, actually) with whatever your shooting,
You want a nice fast shutter speed to give you a sharp picture. This also helps if the animal your shooting doesnt want to stay put for too long! Ermm Try to have a contrasing background, like in the 1st picture there isnt much contrast between the dogs fur and the carpet. But in the second the dark grey capret looks nice behind the white fur.
Ermm Try to be near natural light, obviously outside is the best(its all about light in photography really!) If not then near a window. You can also try and overexpose the Picture a little bit because the fur more likely absorbs alot of light and with it, detail.