Bear in mind that Christmas is peak period so things are more expensive then. If your kids aren't at school yet you mght do better to avoid the rush and go at some cheaper time.
Also, it's fairly important that kids stay quiet while on safari otherwise it may scare the animals away.
My own preference is for Kenya and Tanzania, where animals are fairly easy to spot because it's mostly rolling grassland. Further south in SAfrica or Zimbabwe (if you want to go to Zimbabwe at the moment) it's bushier and you seem to spend more time being driven around without actually seeing anything. But from your kids' point of view that still may be fun. Generally you go out being driven around early morning and late evening when it's cooler, as the animals sleep during the heat of the day. Some places do walking safaris, which gives you the chance to study elephant dung and dung beetles up close!