I'm not pulling up the manual page for you. I'm no
@Boarderguy...
LOL!
But, I think it says somewhere in the manual that, when it's turned off it checks ambient pressure once per minute. That is how it knows to turn on and go into dive mode automatically (and why the battery still runs down gradually, even when stored). So, yeah, it could take a minute to go into dive mode, even at 10'.
If you are really concerned that your Shearwater might be defective and not go into dive mode when it's supposed to, tie a string to it and let it hang down in the pool for a couple of minutes then pull it up. It should be in Dive mode and you should not be able to access "Turn off" until you either wait for it to end the dive automatically or you choose "End dive".
Anyway... I know it's hard! Seriously, I do! But, try to chill out and wait for your class. The class is so you can learn about all the equipment and how to use it. There is not really a reason to worry about that ahead of time. Trust your instructor, relax, and, most importantly, don't go spending a big pile of money on stuff before you've taken the class.
It IS possible you will learn something in class to help you know what will best for you to spend your money on....