^What he said. Also, I found it helpful to keep your lungs as full as you can. When you're not wearing the helmet, lean your head back a bit to force your chest up (helps utilize the buoyancy of your lungs).
When you've got the helmet on, lower your head into the water "vertically" The helmet will trap some air and will support a good portion of your weight.
Finally, pay attention to the videos and instruction, and LISTEN TO THE INSTRUCTORS. They are not there to weed you out - they are there to train you and get you up to USN standards. They will bust their butts to get you where you need to be. API swimming shouldn't be anything to worry about - it should be a nice break from the stuffy classroom.