You mentioned that you have a 1 year old little girl who gets curious while you're changing her diaper and she touches her private parts, and then (even more fun) she grabs her diaper out from underneath her, and when it's a dirty diaper, that's just a lot of fun for you both. So I do have some recommendations for you. First and foremost, hopefully you'll find it helpful to know that this type of behavior is normal. Toddlers will explore all areas of their body, and a lot of psychologists say that that's not something to be discouraged - it's a part of normal sexual human development, and it just needs to be done in a healthy manner. And so don't freak out when she does it. It is totally normal, but you can try to distract her.
Turn on a song, make it light-hearted, put something in her hands that is going to keep her hands busy and keep her mind distracted, and hopefully that will make changing her diaper easier for you both. Also put a pad underneath her so that if she does do it, then it doesn't create such a big mess. And when she does it (when she pulls the diaper out from underneath of her), firmly tell her "No", but don't overreact, because sometimes that causes more negative effects - the child does it more to get attention. So just firmly tell her "No", move on from it, keep her distracted, and hopefully changing her diaper will be easier for you. If you have more questions for me in the future, feel free to ask them on our Facebook page at http://www.facebook.com/IntermountainMoms, and recommend us to your friends and family too.