I've only been twice, back around the turn of the century, but I've kept reading and talking to people. I may go again, maybe not, but you must by all means go if you can. I suggest joining Safari Club International - I'm not connected, just a member, but if an outfitter is on their good-guy list you can be pretty sure he's not fly-by-night, and you should read their publications and go to at least one convention.
You certainly do not have to spend tens of thousands. If you want to spend a lot you can - fivestar accommodations in a prestige area hunting high-end animals, you can even charter you own airplane and bring a professional film crew, but you don't have to. Private ranches set their own rates, and with the global economic downturn and favorable exchange rates, today's prices may be a bargain that doesn't last long.
For plains game a good 30-06 or 308 is plenty, or maybe a 7 or 300 mag if it's wide open spaces. Dangerous will require a 375 or better, something starting with 4 is better if you can handle it, but accurate shot placement is the key. You will almost certainly NOT have to deal with a charging anything. Handgun hunting is done, people do it with archery, but I wouldn't recommend it for the first time. Protecting you is your PH's job and he will be very good at it. When you get there you will realize there is nothing to worry about. Don't try to sneak anything in. I recall a sign I saw long ago near the old Iron Curtain: "Don't become an international incident."