Been on 8 African photo safaris to date so feel free to DM me with any specific questions. Generally, Uganda is mostly visited only for the gorilla trekking as the other countries have better general wildlife viewing, especially the predators. Most people add Uganda at the end of a safari to Tanzania or Kenya. For a first time visitor, East Africa (Kenya or Tanzania) is probably the easiest along with South Africa. Botswana is the most expensive unless you use a mobile camp.

As for packing, my only advise is to not overpack. Most camps will provide some level of laundry services. Depending on the time of the year, make sure to have a warm jacket, etc. as game drives happen early in the morning and evening and with the moving vehicle it gets a lot colder than you would expect. I just returned from Botswana a few weeks ago and even though it was 100 degrees in the day, the mornings were cold and I had to cover with a blanket as I only packed thinner shirts. I would not worry about health or safety. You will need malaria pills for some places. Some also require yellow fever shot but I have not had or met anyone that had issues. Most places require or recommend travel insurance in case something happens healthwise.

As for travel from the US, the best Africa connections seem to be via the Middle East (Qatar Airways, Emirates, etc.) Ethiopian Air also has good prices/connections although I am not a fan of their airport.