I've been travelling throughout Africa for more than half of my life, I started at 18 and now I'm 37... I can tell it's the continent I love most. I'd answer such an open question pointing out a list of things you should really do and things you shouldn't. This is just a list. Each of this points fully deserves a wider coverage. The order in the list is just casual:

things to do in Africa

1) cross the Sahara Desert in Southern Libya or Southern Egypt or Algeria, the best places to feel the great emptiness, lonelyness and vastity of the mighty Sahara and its changing landscapes.
2) go to Ethiopian South Omo Valley to meet people who live following a lifestyle which goes back to 5000 years ago
3) Cruise on the Congo River from Kisangani to Kinshasa on a baleniere, through the heart of darkness
4) go hunting and gathering with the Hazdabe bushmen in Lake Eyasi area, Tanzania
5) Cross the Danakil depression in Ethiopia, where you can really see that Mother Earth is alive and kicking
6) visit Southern Serengeti between January and March, where millions wildebeests gather in the short grass plains to give birth to their calves, followed by an amazing bunch of cats of predators.
7) Trek mountain gorillas in Rwanda, Uganda or Eastern Congo and chimps in Uganda
8) sail on a dhow from the mainland to Zanzibar, Pemba, Ibo, or, if you're a tough fellow, to Comoros or Madagascar or up to Muscat from mainland East Africa, following the swahili and shirazi traders' routes
9) explore Southern Angola, where the desert meets the Ocean and the steep dunes end up in the rough sea, forming the amazing lagoons of Baia dos Tigres
10) dive around Mnemba Atoll, Tanzania
11) explore the Swahili Coast from Northern Kenya to Ilha de Mozambique, to feel the weight of the history
12) hike to the summit of Oldonyo Leng'ai (Northern Tanzania) or Erta Ale (Ethiopia) or any other active volcano
13) go up to Northern Cameroon following the market days held at different villages by different tribes
14) explore the Okawango Delta and Chobe (Botswana) during dry season
15) jump on the train from Cuamba to Nampula (Northern Mozambique): it's the most incredible moving market I've seen
16) overland coast to coast at different latitudes, try to keep away from the main roads (even if gravel or dirt) all the time, just go wild and camp out in the bush as much as possible

Things not to do:
1) skip any African big city: from Jo'burg to Nairobi to Kinshasa to Lagos they're nothing else than shit holes
2) don't go to Nigeria
3) avoid both official armies and militias, expecially in places like Congo, Southern Sudan or Somalia
4) Despite beautiful, the classic Imperial Cities in Morocco are tourist traps, avoid hookers, carpet sellers and guides, there as well as in Egypt. These guys are smarter than the Kashmiri in India, and that's really saying something
5) do not go to Kiwengwa beach in Zanaibar (the rest of the island is ok, but just skip this stretch of beach): it's full of Italian package tourists and now the second language after Kiswahili is Italian there.
6) do not go to Malindi, another Italian enclave, nor to the beaches South of Mombasa, a Germans' favourite
7) refuse day trips to the desert organised from Tunisian Coastal resorts: they're a scam, as well as the "night under a bedouin tent" arranged by agencies in Sharm or Dahab. go to Algeria, Libya or Egypt to see the real Sahara and meet the rwal Touaregs
8) always pay attention to food and drinks: what a local and even a long-stayer can eat and drink is not for the newcomer. But if you've been to India Africa is a clean place
9) avoid malaria, always sleep under a net
10) ask for advice when camping in the bush, expecially in and around national park. Avoid waken up by the light touch of an elephant trunk, by a hippo roaming around the camp, or, even worse, by a bunch of hyenas attracted by your smelly hiking boots

These are the very first things which came to my mind. Africa is so great that the do's list could go on...