Last time I went (2018) I contacted my outfitter to arrange an electronic funds transfer. As previously mentioned you will need your outfitter's bank numbers.
What I haven't seen mentioned is the fore warning: Which your bank can/should help you navigate around: whether sending one lump sum or multiple payments (never the same amount twice): you will need your outfitter's business name and owner/operator name, address, and local (overseas) outfitter's phone number, and reason for the transfer of funds to an overseas account.
You will send off alarms and wave red flags to the alphabet government agencies, who will first assume you could be supporting foreign terrorists. Then these agencies will run checks on the information you provided, check to see if your outfitter is legitimate and that you are legitimately going on safaris or vacationing abroad.
Be mindful that you can not legally leave the US with more than $10,000.00USD or excess amount of foreign currency without declaring the amount(s) and intended purpose.
It is also note worthy that you also have CBP 4457's for high dollar electronic equipment [ie. phone(s), laptop(s), etc], jewelry, etc. to avoid paying import taxes upon return to the US. Any shopping done in the Duty Free shops; retain and have handy your sales receipts.
Be mindful on what items you can and can not import/export into/from the US or export from foreign countries.
Since this is your first trip, If you are traveling with your own weapons, I and others here on AH would recommend you use a preapproval firearms permit agency such as Henry Rifle Permits to assist in completing SAPS 520 and getting your firearms through SAPS, providing you are hunting in or transiting through South Africa.
Check out the posts in Before and After the Hunt for more helpful information.