Depends. Some boards come with 'channels' where fins aren't needed. A board with channels may not even have holes to screw in fins. A board without channels or fins is very slippery and harder to maintain stability. However, it can be great for doing surface spin tricks as the lack of fins provides easy spinning.
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Fins allow you to propel yourself much faster than without fins thus allowing you to catch a wave much more easily than without. In bodyboarding using your legs to propel yourself is the primary means of propulsion to catch a wave as opposed to using your upper body as is required in Surfing.
Swim fins are not necessary for most swimming lessons. Lessons are originally meant to teach you how to swim correctly, which means paddling your arms and feet. In a true emergency, you probably wouldn't be wearing fins, so it's effective to know the correct techniques.