I echo re Botswana - we're thinking of going back there next year. There's such a good mix of habitats which broadens the opportunities for what you can see. It's a long time since we were last there, but I do remember it being quite pricey.
Zambia is also very good, and you have the opportunity to visit Victoria Falls (although time of year/rainy v dry season dictates how impressive they are).
Someone mentioned you'd be in Africa for the migration season, in which case Kenya or Tananzia might be good options.
I think all of the above are malarial though so you'd need to be taking the anti-malaria tablets. Usually it's one week before you arrive in the malarial area, then all the time you are there, then for I think 2 or 4 weeks after. It does depend though on which tablet is recommended.
If you want to avoid having to take anti-malarials, then one option might be South Africa. The reserves along the east side of the Cape are malaria free I think (haven't actually been to any of them so can't directly recommend). And it would fit well with your idea of a city + safari. You could for e.g. fly London to Jo'burg, have a day there with a driver to take you around (the Aparthied museum is a must-see (although not sure whether youngish kids would enjoy it?)), then take a flight down to Port Elizabeth, pick up a hire car and drive up the Garden Route (or have a driver + minivan to do the driving for you) and finish up in Cape Town. You could visit a couple of game reserves along the way for a few nights each.
On the garden route there's also lots of good places to stop overnight and you'd be there in whale-watching season. In and around Cape Town you've got the Cape of Good Hope; the beach where all the penguins are (can't remember what it's called); Table Mountain; Robben Island. There's also beaches further north.
But it's August, so you'd need to check the weather - it might not be that great.
Definitely contact a few agents to pick their brains, see what they suggest for your budget, time of year and, of course, for something that is child-friendly.
These are some companies we've either booked with or spoken to in the past and all have been on top of their game :
Explore unforgettable African safari tours with Pulse Africa. Luxury Africa trips, tailor made safaris, and once-in-a-lifetime wildlife adventures await.
pulseafrica.com
Your African safari starts with Expert Africa. We're an award-winning travel company specialising in high-quality, tailor-made safaris to East and Southern Africa.
www.expertafrica.com
But there are loads of others out there.
A few tips/observations if you do go for the safari option :
- Try to visit at least 2 different reserves in your trip.
- Make sure you are in a reserve where the vehicles can drive off-road.
- Check how many people the camp generally has in each vehicle - the fewer the better in terms of being able to have a clear view of what's around you, plus being able to hear your guide (although there is an argument that more eyes are better for spotting wildlife).
- Be prepared for early morning starts - usually wake-up calls around 4.30-5.00am, a cup of tea/coffee, a bit of toast then you're out around sunrise for your morning drive, with breakfast when you get back. Then there's lots of downtime until you go out again around 4pm ish, back after sunset for dinner. A camp that has a watering hole increases your chances of seeing things during the down-time.
- There's often the opportunity to visit a local village or school etc. Not really our thing, it always feels a bit exploitative, but at the same time it does bring in money for the locals.
- Buy, or find someone who can lend you, some binoculars, if you don't already have them. They are essential.
I'll stop there as I could go on for ever on my favourite past-time of holiday planning! Do contact a few agents to get some ideas.
Feel free to DM if you need any more info.
Good luck!