Samsung removed the Bixby key on the Galaxy Note 10 and Note 10+ and gave the power button dual functionality. In the process, they changed the way you power off the phone and made taking screenshots a bit tricky. Fortunately, it's not too hard once you get used to things.
By removing the Bixby key while keeping Bixby itself, Samsung was forced into making the power button perform multiple tasks. When you long-press the key, Bixby opens by default. So now, accessing the power menu requires two buttons: power and volume down. But these are the same two buttons you normally take a screenshot with, hence the confusion.
To take a screenshot on the Galaxy Note 10 & 10+, you need to press both the volume down and power buttons quickly. Unlike other Android phones, you are no longer allowed to hold the buttons down (doing so will actually bring up the power menu). I can't emphasize enough how quick this touch is. Press both buttons together, immediately let go, and you will capture an image of your display.
Depending on your preference, you might end up using the gesture instead. Built into the Galaxy Note 10's One UI is a gesture for taking screenshots. With the side of your hand, swipe left or right to take a screenshot. Make sure the side of your hand is touching the display as you move from left to right and the screenshot will automatically be taken.
Another way to take a screenshot is to use the S Pen. Simply take out the stylus and select the floating action button to pull up its menu. Choose "Screen write," and you'll have your screenshot.
Finally, you can always use Bixby. If you set up Bixby to wake up as soon as you say "Hi Bixby," you can use voice commands to take a screenshot. To initiate, just say "Hi Bixby, take a screenshot."