How to take a good picture of an animal or person?

First: i have a Nikon D3000, i love taking portrait's of animals and people, i have an issue with blur and a dark or overly bright photo, my goal is to get a nice clean face picture with a blurred background. Also i have another issue with taking photos in a dimly lighted room, i slow the shutter speed down to get enough light in, but if the person moves just a little bit they come out blurry. So please help, it would be greatly appreciated.

P.S a film user for a long time ( Ricoh xr-p) and just switched to digital

my goal is to get a picture of an animal like this it's goofy i know.
and a portrait like this

Http://www.digital-photography-school.com/how-to-get-better-digital-photos-in-low-light-conditions-without-using-a-flash

http://www.ehow.com/...round.html

Above are some links which should help you.