It doesn't look like my answer posted...so I'll try again...
We waited until my daughter was around 8--old enough to care for a pet. We started with hamsters. (It was all I could handle at the time.) We had a contract with her and explained all the rules and things that needed to be done. We explained that if she missed cleaning the cage 3 times, we would get rid of the hamsters. She did really well, and later we got other pets...when I was ready.
Let me say that if you are considering a dog, I think you will end up being the primary caregiver. I don't think there's any way around it. Puppies, especially, are a LOT of work. It's like having another baby in the house! Everything goes in the mouth and you have to watch them all the time so they don't mess on the carpet. You could adopt an older dog, but it still is work. Someone has to let them out on schedule. Sometimes they get into mud or whatever and you have to give unplanned baths.
Cats are really pretty easy and don't require constant attention like dogs, but there is the litter box to clean periodically.