I like that you are keen to try different food and local customs, but as a South African, I can tell you that "game" meat is not such a big deal or favorite for us as you might think. Most South Africans eat the "normal" western meat options, ie beef, pork, lamb, chicken, and only rarely will you find variations on those. We see most game meats as a type of novelty, mostly for tourists, so that could be why you got the funny looks.
Most game meat/venison is quite tough compared to beef. At the same time beef is quite affordable here (compared to Europe and elsewhere), and in very good quality, so most people will rather eat beef or lamb if given the option. There are very few restaurants that I know of where you will be able to order any type of game meat, and then it will usually be 90% antelope (kudu, impala) meat.
With regards to what you can ask for and not - well basically if they say it's game meat, it could be a variety of antelope meats mixed together, and it will probably be in a stew type format. If they specifically say it is crocodile or ostrich or something then you are lucky, and then you know exactly what it is. Just be aware that some places tell you that it's crocodile meat (or something exotic sounding), when it's actually something else like a general mix of game meat.
What I can recommend though is that Ostrich meat has become very popular and even affordable here now. Its a red meat, but has about zero cholesterol, so very healthy and tastes great! You might find this in more restaurants, and can easily pick it up at the super market.
Also a warthog spitbraai (the whole thing slow roasted over a fire) is also amazing and I can highly recommend, but you won't ever find it in a restaurant or such - you will probably need to know someone who does it or be on a game farm where they offer that.
Lastly, you can try biltong from several wild animals, mostly it will be Kudu, impala, gemsbok, blesbok or other antelope varieties.