I would actually suggest that you start with guns and then if you want a greater challenge, try bow hunting, aafter you become more proficient stalking, etc.
The best thing you can do is to get a simple bolt action .22 next. I hear it is great fun and challenging to try and get Guinea Fowl by sneaking up and shooting them in the head. You have various hares, etc. available as well.
Then move up to either a shot gun if you want to pursue more birds, or a larger rifle like maybe a 7x57 or 30-06 for Plains Game.... Then when you are proficient at that, move on to larger and potentially dangerous game or bow hunting the same plains game.
But do yourself the favor of starting small and moving up.... And by all means, if you are more excited about bow hunting than rifle, go right to that... Heck you can even chuck spears or use a slingshot if you want! LOL
Personally I started with a homemade slingshot as a young boy going after rats and English sparrows (air rats), then a bb gun, then a .22 for squirrels and rabbits, shotgun, and then on to whitetail deer with several rifles, and now finally Africa!!!