There is a company that specializes in organizing flights for hunting trips in Africa. Don't remember the name but here's her email kathiwildtravel.net. She will give you all the ins and outs and will make it all simple. She will be familiar with the gun laws/permits and potential problems.

Personally, I'd probably do a plains game package to RSA or Namibia for a first trip, but nothing wrong with slamming a cape buff right off. Zimbabwe is probably the spot for that when considering costs.