I caught my first salmon today but I wasn't too sure how to hold it. What I did was grab it by the tail and cradle the top part of it's belly, but it felt a little weird. Any tips?

You did exactly what you need to do when holding a fish. 

You got it right!

 

...and get your hands wet first.

If he caught it inshore or in a trib, and it's kyped up, won't matter - the fish will die soon.