Getting to Kenya, Tanzania, and Uganda is no problem. All of the major airports - Dar es Salaam, Kilimanjaro, Nairobi and Kampala - have daily direct flights from Europe, usually via Amsterdam or London. Once you arrive, you can usually find flights between smaller destinations like Zanzibar, Lamu, Selous, or can book small private planes to take you to game lodges or other places that are difficult to travel by road.

Precision Air Tanzania
Air Tanzania

Visas
Remember, visa regulations change and it's important to check with your embassy before visiting the country. In East Africa, you can usually get visas upon arrival at the airport, or crossing a border. However, if you're traveling overland through a remote border crossing, like from Mozambique into Tanzania, or even by boat from Malawi, make sure to ask around at a police station to see if it's possible to find someone to give you a visa first.

Tanzania
Visitors coming to Tanzania can get a three month visa upon arrival at the airport. Cost is usually $50, payable in US dollars, but can be cheaper for some of the Commonwealth countries. There are both single and double entry visas available upon arrival. If you're coming to Tanzania via Nairobi Airport, you can get a transit visa for $20, that allows you seven days in the country before you need to cross into a boarding country, usually Uganda or Tanzania. You can also buy visas for entry into Tanzania at the highway crossing from Nairobi into Arusha.

Kenya
Kenyan Tourist Visas are $50. You can get these at the airport or before arriving in the country. If you plan to get a multple transit visa, meaning you are coming into and out of Kenya multiples times, the cost is $100. A transit visa to Tanzania is $20.

Uganda
The Ugandan Embassy in Washington DC states that all visas to Uganda are now $100. Although this seems like a ridiculous visa price, it might be done at a border crossing or at the airport for cheaper.