We have a dog and a cat right now, and a 6 month old daughter.
I never had pets growing up, and I feel like I missed out on the relationship a lot of kids have with their pets -- most of my friends growing up had either a dog or a cat and they really were wonderful friends and companions.
I can already see the bond that is developing between my daughter and the pets we have. If she's upset for some reason, both the dog and the cat will run over to try and see what's wrong or comfort her. She seems to be soothed a lot by their presence too - when they are in her line of sight her face brightens up so much!
Did you have a pet as a child? If you have kids now, do they have a pet as a companion? What is their relationship dynamic like?