Generally trademark rights are territory by territory, but any trademark owner can use the "Madrid Protocol" treaty to apply for registrations in multiple countries that are signatory to the treaty, including the countries you've specified. You can file an "International application," which gets submitted to the countries you've chosen, and each member country applies their own rules to determine whether or not the mark may be protected in their jurisdiction.
See an IP lawyer for help.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.
You ask: Can I just register it in South Africa and still be covered or at least stake a claim in UK, USA, AU and CA?
Response: No. Your South African registration would be valid only in South Africa and the only way to "stake a claim" for trademark rights anywhere else would be to apply to register the mark in the other country and/or use the mark in the other country. As noted, there's a procedural mechanism that assists people who want to register in more than one country. See http://teasi.uspto.gov/common/madridfaqs.html
You ask: If I get a trademark only in South Africa does this mean anyone else can register it in a different country?
The only reason to seek trademark registration protection in more than country is because you're doing business in more than one country. And if that's the case, then you need a business attorney to help you navigate those waters. That business attorney will call in a trademark attorney when he or she needs trademark advice. Establishing international trademark protection is WAY over your head. Speak to a business attorney.
While I do not disagree with the other answers posted thus far, what you should do may depend on the reasons you feel you need to be covered in the other countries you listed. If you're only interested in the ability to stake a claim, you may be able to do so without trying to obtain trademark registrations (in some common law jurisdictions, you get some limited rights based on your commercial use of the mark even in the absence of a registration). If you don't expect to actually use the mark in those other countries in the short term, you may want to seek registration based on your application/registration in South Africa, if that is where your business is located. The International Registration process available under the Madrid System can save some money in the short run, but has some limitations. So, the lawyerly answer as usual is "it depends".
Disclaimer: This answer is not intended as legal advice and no attorney-client relationship is created by this communication.