Every family is different, so it really comes down to the type of pet that fits;
We started our kids out with a fish as their first pet. Simple, not to expensive and taught them responsiblity. We have since added a cat and a dog to the mix and all work really well for our family. We have resources available to help look after our pets when we travel and a big backyard for the dog to have plenty of space to play.
Pets are as unique as the families they join- have a really good talk about the 3 factors (Lifestyle, Space,Finances) and see what works for your family.
Dogs seem to be a good choice. They have the energy to keep up with kids (and an interest in playing with them in the first place) and the rest of the family will become just as attached to them. They can be taken on holidays and on family outings (in fact, they can be the reason families go out together, like on walks). They're the kind of pet that becomes part of the family as everyone grows up together. You need to be ready for such a demanding pet though. Fish can be easier, but they're not as exciting and die too frequently.
I think as a first pet, maybe goldfish. To teach kids the responsibility and as they get older other pets. It really depends on the family though, where they live and their lifestyle. Small pets for those who don't have as much space and dogs and cats for homes with larger areas.