Yes, you will need a hat and sunscreen and a change (or two) of clothes. You have to trust me on this - no one studies your outfits while you are on safari. Really. Eagles, zebras, lions, Maasai warriors and spectacular scenery draw all the attention.
If you are not stopping in Europe you can pack everything into a single duffle bag. This bag plus a carry-on for cameras, medications, and your field-guide-to-the-birds is all you will need. If you are stopping in Europe you should pack all the Europe-stuff in a suitcase that can be left behind and retrieved later. On safari there is little room for storage and transport of luggage in the vehicles. One duffle per person works pretty well. Luggage beyond that and luggage that is hard-sided make packing very difficult for the drivers. The space behind the rear seat will have a case of bottled water and a spare tire (or two) in addition to the luggage.
If there are four or five people in your vehicle the storage space will get filled up very quickly. If you are arranging your own safari it is always a good thing to plan for the people, the itinerary, and the luggage. Seven people and their luggage is a crowd in a single vehicle. Four people and their luggage is a breeze. The cost of having another vehicle, driver, and related arrangements does impact the bottom line; but consider the impact of crowding on travel, photography, and ease of observing the wildlife. For ten, eleven, or twelve people two vehicles is essential. The same can be said for seven or eight people. Once you get to the six and seven participant level you have some hard decisions to make regarding how to divide your group.
As far as clothing goes there are a few considerations. The weather will be warm and possibly hot. You may never wear a fleece, sweater, or jacket. Wool and heavy cotton are often too much. Sandals work well on most days, as you are inside a vehicle. Outdoor sandals can get you through the whole trip; especially those with a protected toe. You will not need leather shoes or dress clothes. Sneakers (or today's equivalent) are fine.
It is easy enough to bring small numbers of wash and wear shirts, pants, and undies. This kind of clothing will dry overnight or during nap time in the African sunshine.
Pants - wear one and pack two. Shirts, maybe pack three or four (and wear one); but remember, the safari lodges will sell shirts and T-shirts if you need more. Most people say that blue jeans are too heavy. Undies can be of a poly fiber that can be washed and dried with ease. A couple pair may be boring but certainly adequate. Poly fibers and other man-made fibers tend to be thinner and dry much more quickly than does cotton.
Here are some further thoughts on clothing; a bathing suit is useable at most lodges for those with the inclination and energy. A fleece, hat, or even gloves might used one or two early mornings if you are visiting in June, July, or August. Some people bring a face mask to filter out dust. This may be a reasonable idea for people with respiratory problems. Dust is possible but quite variable. Some years it is mud not dust. Bug spray is rarely needed and along with sun screen, mouth wash, and all toiletries can be purchased en route.
In summary you should pack at home - then remove at least one-third of your choices. The less you pack the easier the storage, the more room for trinkets and souvenirs, and especially, it leaves more room for binoculars, diaries, and cameras.