Easiest airport to fly into will be Mombasa, departure will be Nairobi. You may need to transit in Nairobi depending on where you're flying from.
Yes, you'll need an Electronic Travel Authorization. Luckily, the application is easy and can be completed online. Just plan to do it a few weeks in advance. More information can be found here: https://immigration.ecitizen.go.ke/index.php?id=5
Please check with your doctor or a travel health clinic for personalized advice. You should definitely be up-to-date on all routine vaccinations and yellow fever is typically recommended.
There are instances of malaria at the Coast as well as some of the national parks (not Nairobi though). Prophylactic malaria pills can help minimize risk of infection.
Tuk tuks!