The most common (at least for foreign tourists) is to book into an all-inclusive "safari lodge" and have a first class, once-in-a-lifetime experience. We've all seen these in movies - glamping in a five star resort with champagne and caviar. You sleep in a tent in a platform in the trees, and are waited on hand and foot by staff who outnumber the guests. These places are unbelievable, and cost $1000-3000 per day per person. As you can imagine, that's not what we're doing.
The second option is to do-it-yourself. You fly in, rent a car, drive cross-country to the park (400 km from Johannesburg to Kruger), get a room in or near the park, get up at 5 am, then drive yourself around. Given that we're here during South African summer vacation, there are lots of middle class families doing this here this week - and one Canadian family.
If you're in a private park, the guides drive you over hill and dale to find animals for you to see up close. Given that these places are located on private property, the drivers can go wherever they want (on or off road), and can go out after dark when the big predators are active. This pretty much guarantees up-close experiences with big game every day. For those wanting to see big cats, this is especially important.
On a self-driven tour of a national park, you can enter at 5:30 am and must be out by 6:30 pm. You must also stay on the roads (paved and gravel) at all times, and spot the animals yourself. As you can imagine, if you were doing this along Highway 60 in Algonquin Park, you wouldn't see many wild creatures. Here, however, the density of animals is such that you have a much better chance of sightings.
As such, budget safari makers in Africa only see animals that are alongside or crossing the road. But don't feel sorry for us! Here is a sample of what we saw in our first couple of days.