When you are first starting out with freediving, I recommend you get a pair of tough rubber fins like the ones below. They are easy to learn with, and training with them is the best way to develop good finning technique.
What's good about fins like these
Good for learning
If you are finning incorrectly, these fins won't do the work for you. You will get instant feedback to improve your technique.
Strappy scuba fins are ok to rent for your first ocean day, but if you are going to buy fins, make sure they have a full foot pocket, to efficiently transfer the force of your kick to the water.
Blades are removable
The blades can easily be removed from the footpockets by unscrewing the two screws in the backing plate. This means that you can upgrade to carbon blades when you are ready without having to buy new footpockets.
Fit in a carry-on bag
You can train with them in the pool
What to avoid when buying fins
Straps on the back
Scuba divers often use fins with straps on the back, to fit around their boots. This kind of fin does not efficiently transfer the force of your kick to the water, wasting energy and oxygen.
Paying too much
There is a "dead zone" of $50 - 150 in the freediving fin market where a lot of older models are priced. These include fiberglass fins, huge plastic fins, and scuba fins with weird contouring that look like they have been developed by NASA to withstand re-entry into the earth's atmosphere. If you want to upgrade from training fins, skip all these and go straight to carbon fins (see below).
Carbon fins offer much more power than any other type of fin. Carbon has 100% "memory" so it snaps back to its original position with no loss of energy. You don't need carbon fins when you first start freediving, but if you have been freediving for a while and are reaching depths from which it is hard to swim up, this type of fin starts to come into its own.
Photon Fins are an affordable carbon freediving fin. Their blades are 100% carbon (no fiberglass) and 550 micrometers thick at the tip, which is a medium stiffness appropriate for our conditions. They use the new style of "no rail" footpockets, which allows the carbon blades to flex along their whole length.
Top of the line: Alchemy Fins
545 plus shipping
Buy from Alchemy
One of the most popular competitive carbon freediving fins. Alchemy's carbon fiber has a complex construction, with different hardnesses along different parts of the blade.