Welcome to AH Forum!
As a first timer the planning can seem daunting but it is actually half the fun of the hunt. There are some specifics you need to be aware of on the travel and gun imports but study will make them clear. I would like to suggest you read my current post "Just Back from Namib with Jamy traut Safaris". I have detailed exactly what is required when you get to South Africa or Namibia with your weapons. Hopefully it will be of some help to you.
As to where to go? South Africa is the easiest to get to (especially if you pick a place with in 2 or 3 hours of Jo'burg) and possibly cheaper in the long run because of that. As my last trip 1 week ago was my 5th, I can recommend a couple of SA outfits. Sadaka Safaris is about a 2 hr drive from the airport. I went there last year with 4 other first timers all of whom are rebooking again because they had so much fun. Another SA outfit to consider is Xomaqua Safaris. Both searchable on the net. Xomaqua is a little futher from Jo'berg but a fine outfit to hunt with. Please do your research before booking and understand that South Africa is set up for a lot of high fence hunting but usually on very big properties. Not the small places we see in the USA for high fence hunting. You still have to hunt them even if there is a fence somewhere distant. Understand that if it wasn't for high fence properties there probably wouldn't be the hunting industry or animals in SA that you see today. If you want to travel a little further go to Namibia. This is the least populated country in all of the African continent. Miles and miles of nothing but miles and miles of bush country. The animal species are somewhat more limited but for your fist plains game hunt it will not be a problem. Kudu, zebra (both kinds), blesbuck, black and blue wildebeest, heartebeest, springbok, eland, steenbuck, impala (limited), gemsbuck (by the hundreds) and others are all there. My recomendation is jamy Traut Safaris at the Panorama Camp (please read my report). You can't go wrong there. Your "friend" will love it. You have no idea what you are in for as far a service goes when on a good safari today. You sleep on clean sheets, you have a full en suite bathing facility, laundry is done every day (don't pack too many clothes, you or she won't need them), you eat with crystal and linen every meal (no sitting on a cold log in the snow eating beans out of a can), you have a chef preparing the meals and you shoot and take pictures, everything else is done for you.
You will find all the help you need right here on this forum. Many of the commentators have much more experience than I do so ask away on all your questions. This is the best forum I have ever seen on any subject. These folks are here to help you enjoy your first safari. Read all you can here on preparations. Lots of good info available. Search my name 'cliffy" for a couple of other articles on how to do things. Just remember-
IF YOU GO ONCE-YOU WILL RETURN-DEAL WITH IT!