Don't you think you should go thru the entire greiving process first? Getting another pet right away is a mental crutch to avoid mental pain and suffering.
I didn't have pets growning up, and after my very 1st dog died(when I was 34 and I hadn't had a child yet), I'm SO glad I waited a couple months to get another dog.
No, get another dog right away, it heals the hurt. I am speaking from experience, I have owned and lost dozens of dogs. A dearly beloved dog can be replaced, the old one won't be forgotten (I still miss Tootie) but you have the new one (Cookie) to love and focus your attention on. I have lost a child too and there is no comparison in the experiences. A child can't be replaced and you must spend time grieving. People who try to put it behind them with drugs and alcohol or another relationship are using a crutch and they are just fooling themselves.