Watching animals is also one of my biggest hobbies. While I've seen many animals in N. America and Europe, little compares to a trip to Africa. I recently returned from my 3rd trip to southern Africa and I can recommend some places. Tourism is booming in S. Africa and it seems like private preserves are popping up all over the place. They are expensive and too cushy for my tastes, plus I would feel like I'm in a driving zoo of sorts.
For an easy do it yourself driving safari in S. Africa one of the best places is Kruger National Park. It is a very large park (~ 2 million hectares) with an abundance of wildlife (elephant, lion, leopard, cheetah, white rhino, buffalo, hyaena, baboons, etc.) including endangered species that are difficult to find in many African parks (black rhino, African hunting dog). The birdlife in Kruger is also superb.
In terms of walking safaris, you do not typically see as much as when driving , but they are a completely different experience and much more exciting. They are still relatively affordable in Kruger compared to other parks.. They last 3 days and are quite amazing. We saw a hyaena eating an impala (after stealing it from a leopard), elephants, giraffe, white rhino, lions, wildebeest, crocodiles, and a variety of bovids (impala, kudu, etc.). The highlight, however, was getting charged by a black rhino. Thankfully, the guides deterred the rhino from finishing the charge with a warning shot. The entire incident got my blood pumping.
S. Africa also has fantastic backpacking and peakbagging opportunities in the Drakensberg Mtns - the inspiration for the mountains in Lord of the Rings. I went on a 4 day backpacking trip there to visit 5 caves full of Bushmen rock art and I will definitely head back to the area next time I'm in S. Africa. Although I have only climbed a bit near Johannesburg, good rock climbing exists in S. Africa, especially the longer routes down near Cape Town.
Otherwise, if you would like a more remote safari I would recommend heading to the Okavango Delta in Botswana. It is a much more private experience and there is nothing like knowing that there are hyaenas and lions outside the tent (bring a pee bottle!). We had a territorial battle at our tent site between 2 hyaena packs. Three lions eating a Cape Buffalo was a great sight there as well. You will see many of the same species as at Kruger, but with Homo sapiens sightings much more rarely.
You will see fewer people on safari in southern Africa than in E. Africa based on my conversations with people that have traveled to both areas.