Americans do not need a visa to travel for up to three months in Botswana, Namibia, or South Africa.
Americans DO need a visa for travel to Zambia. Visas can be purchased at the port of entry or in advance. A multiple-entry visa for Zambia may be obtained through the Zambian Embassy in Washington for about $40. If your visit to Zambia will only last one day, a day visa is easy to obtain at border crossings such as the one in Livingstone.