I’ve been several times, mostly recently last summer. It is truly a magical experience that you will thoroughly enjoy.
South Africa is about as far away from Washington as you can get, and you’ll be in the air about 20 hours plus or minus. If you are taking firearms, the cities you connect through should be a consideration. I’ve always flown Delta through Atlanta because they generally do a good job with firearms. Some hubs such as London, Amsterdam, and others have reports of higher rates of poor experiences for travelers with firearms.
Of course you can avoid all those hassles by borrowing a firearm from your outfitter, which I did last year for the first time.
Try to get there on no more than two flights. Every added connection creates more potential for problems.
If you take your own firearms, I would advise using a service such as Rifle Permits.com. Getting yourself messed up with the South African police because you filled out the form wrong, and you’re tired and cranky from 30 hours of travel isn’t a good way to start out.
Best wishes; I envy you.