Firstly, do not be obligated to purchase a bunch of new gear! My Photo Safaris can assist you with equipment rentals to lesson the load and risk you need to travel with. Visit our Equipment Rentals page for more details.
Typically on a photo safari you will achieve good results with a telephoto lens of 300mm or more, a 100-400mm zoom and even a 70-200mm are great lenses coupled with extenders. A wide-angle lens for scenic and scapes shots and a spare body and macro lens will add to your arsenal.
A good camera bag is essential and one that can endure some dust and perhaps a little rain if (most good bags have a built in rain jacket). Hard cases (Pelican) are also good but you must consider your baggage restrictions.
Memory cards, you can never have enough! It is also recommended you bring your laptop already loaded with you post processing software (preferably Lightroom and/or Photoshop) with adequate hard drive space and a portable drives as a backup.
Also bring spare camera batteries and a charger, although there is power at the destinations you will never be sorry for carrying an extra camera battery.
Filters are very handy, a polarizing filter will be used often and for landscapes a ND filter will be handy.
A flash is not essential but is some cases it can help as a fill-in flash for shadows in low light.
Tripods, mono pods and beanbags will be supplied by My Photo Safaris.