I'm afraid it's more difficult than you realise. You can certainly stay in various South American or South-East Asian countries. You are almost certainly not allowed to work there, even remotely, but it's probably possible to get away with doing it.
Europe is another matter. The Schengen agreement was intended to make some things easier but it's not particularly helpful for you. First, it means that the maximum stay limit (up to 90 days, depending on your visa) applies to a whole lot of countries, instead of country-by-country. After that, you need to spend a 90-day cooling off period elsewhere. So it's more difficult to travel indefinitely or take a long time to enjoy each country without falling foul of the regulation.
Additionally, as a South African citizen, you would need a visa. You will have to obtain this visa from your usual place of residence, which if you are constantly travelling probably means South Africa (as you won't be able to secure a proper residence permit in other countries while traveling on tourist visas or visa exemptions). So you can't get it on the road or spontaneously decide to go to the Schengen area.
An additional issue is that you need to satisfy the consulate that you have a legitimate purpose for your trip and will return to your country of residence after that. Here, being able to work remotely works against you. They most definitely don't want you to work while in the Schengen area (that's not a legitimate purpose if you don't have authorisation) and prefer to see tangible things tying you with South Africa (which is more difficult with a nomadic lifestyle).
If your citizenship would allow you to visit without a visa, you could possibly get away with it a few times (although the authorities are trying to fight this) but since you need a visa, you will have to submit a lot of information about your situation in advance and have a plausible plan, which makes it more difficult to flaunt the rules.
You also risk getting a visa that only covers the plan you submitted, which might not afford much flexibility. And if you rely on your job to satisfy the “means of subsistence” requirement, you can't very much submit a plan for a three-month tour of Europe without disclosing that you intend to work remotely. But if you ask a visa for a trip that corresponds to, say, a couple of weeks paid leave, you might get a visa valid for 20 or 30 days only, which is very short.
So that's a large part of Western Europe that can't be visited without extensive planning and many administrative formalities, sadly. The UK (not Schengen) is similar and the US and Canada probably too.
One tip would be to start with the Schengen area and other picky places. You can apply for the first visa while you still have a situation in South Africa and once you have used several such visas (even from different countries), getting another one for subsequent trips should become easier.