For permits to transport your firearms and ammunition out of Canada and then back in, there are a few forms that must be completed from Canada, and you must identify the firearm, and all non-permanent attachments such as the bases, rings and scope, (bipod if you have or are taking one) and if so equipped the detachable magazine(s), and the ammunition, including the number of cartridges. These take approx. 6 weeks to process, so submit well ahead of your trip to have your permits for the trip.
You will also need to have your PAL.
For hunting and taking your firearms to the US, it is fairly simple. But depending on the State where you will be hunting, you will need to contact the outfitter to determine what the specific requirements of the State(s) where you will be travelling to and hunting. While I did not take firearms to Colorado last fall (archery hunt), we had to have a hunter training course and proof of having that training with us to obtain our hunting licenses/tags, and on our persons during the hunt.
For taking your firearm to South Africa, you will need to obtain the required permits for your firearms and ammunition for importing into SA. You will also need to complete the forms and submit your invitation to hunt and contract with your outfitter/PH so they know who and where you'll be hunting with. Again, these take weeks to obtain, so submit early so you have the permits in place for the trip. Many outfitters/PHs will assist or direct you to work with a consultant that will assist with this process, and even meet you at the airport in Johannesburg to assist getting your firearms through customs and to your outfitter/PH. Ensure you get the proper forms for Canada, as many SA consultants (travel and/or firearms) will provide the US forms as standard issue, and these will be no good for Canadians. Also double check on the requirements for arriving in your destination country, and the countries you will travel through to get there, as they also have requirements for transporting firearms, ammunitions, and hunting gear including knives and broadheads. This will include how they must be packed, travel cases locked, documentation within the travel cases, and number items (incl/ cartridges) and/or weight restrictions that must be adhered to, so they will not be confiscated, and/or you fined or worse (detained/jailed???)
Depending on your travel arrangements and airline you travel with, you may also need additional permits for travelling through other countries to/from SA, such as Amsterdam, where you must also have permits for knives. When we went to SA last year, the only airline option we had was Royal Dutch Airlines and had to fly through Amsterdam with firearms, etc. Be sure you research the airlines prior to booking, or using a travel agent not familiar with booking trips for hunters travelling with firearms and hunting gear. There will be certain things you must do to travel with these items, and they all differ to some degree.
For taking firearms to New Zealand, you must obtain the permits for your firearms and ammunition from NZ too, and submit well ahead of your trip, as again, it takes weeks for the permits to be processed. You must also provide info as to where your firearms will be stored during their time in NZ, and where you will be hunting. And if hunting with an outfitter, you must have all their info and an invitation outlining where and when.
This what I learned over the past few years, and hope it helps you and answers some of your questions.