Mask and Snorkle
You will most likely need a mask and a snorkle. Freediving masks usually have lower airvolume compared to scuba masks. This means that the glass is closer to your face to minimize the airvolume between mask and face. This is crucial because you will have to equalize your mask while going deeper and deeper. Otherwise, it will suck onto you face and blood vessels in your eyes run the risk to burst. The smaller the airvolume is the less air you will need to equalize your mask. But no worries, while freediving usually your mask will "automatically" be equalized when releasing you nose after a normal equalisation.
What most likely also should not be missing in your freediving equipment is a wetsuit. A wetsuit will help you to stay warm which is crucial in freediving. Freediving is all about relaxation and lowering you heart rate. Thus, your body will loose a lot of heat without a proper wetsuit. Freediving wetsuits are quite different than scuba wetsuits. They are in general more flexible to allow better movement and are more durable against high preassure. There are also a lot of neoprene options and a lot of lining (inside and/or outside) options.
The inside lining can either be a nylon lining or no lining (open cell neopren). The benefit of open cell neoprene on the inside is that it will stick more to your skin. This will make it harder for water to enter you wetsuit and will further improve the thermal properties of you wetsuit. Downside is, that because of sticking more to your skin it will make it quite difficult to don the wetsuit. Often a form of lubricant (soap or water) is neccessary to properly don the wetsuit. Another downside is, that the wetsuit looses stability and durability without lining which increases the risk of tearing it apart.
The outside lining again can either be a nylon lining or no lining (open cell neoprene or also called supersmooth neoprene). Supersmooth neoprene will decrease friction in the water. The effect is not too high and thus mainly used in competitive environments. Especially because of the already mentioned downsides. The neoprene becomes less stable and durable meaning you have to be very careful to not hit any rocks or other sharp objects as this will easily cut your neoprene.
In my humble opinion open cell neopren on the inside and nylon lining on the outside is a good start. You will have the improved thermal properties but still a decent durable wetsuit.
Fins are also a core part of your freediving equipment. Freediving fins are longer than usual scuba diving fins. Main reason for that is efficiency. As you are very concerned about your oxygen consumption you want to get the most out of every muscle movement. Freediving fins come in different materials and different stiffness. Materials are mainly plastic, fibreglass or carbon and stiffnes ranges from soft over medium to hard.
Plastic fins are the least expensive option but also the least efficient one. After bending of the blades these do not really pop back into their original position and sometimes even stay slightly bended. Fibreglass blades do pop back to their original position and thus are way more efficient but are also more expensive. The most efficient and usually also most expensive blades are carbon blades. The downside of carbon blades however is that they are more fragile than fibreglas. Thus you have to pay a bit more attention and do not really want to hit the reef or anything else with these.
The stiffness between the different manufacturers is not standardized thus it is not possible to give a generall recommendation. As a guideline you can say the taller and heavier you are the stiffer most likely your blades should be. Also your fitness level might influence the appropriate stiffness. For trained swimmers with strong legs stiffer blades might be more suitable. In the end it most likely comes down to personal preference. So in best case you have the opportunity to test the different stiffness to see what feels most comfortable for you.