Woody Woodpecker's screen debut, as others have pointed out, was really an "Andy Panda" cartoon but Woody apparently made such a strong impression that he went on to be a animated movie star.
"Can you really catch a bird by putting salt on its tail? That's the question asked, and how we are introduced to WW. Little Andy had asked the question to his dad, who was busy reading the day's Racing Form (which was filled with puns. Take the time to freeze the picture and you'll get some laughs or groans.)
The rest is kind of juvenile but that had to be the intended audience, at least early on. Also, it's 1940 and humor - especially with cartoons, in my opinion - got much better - by the middle of the ''40s. Early Tom & Jerry cartoons, Daffy Duck, Bugs Bunny, etc., all didn't hit their prime until then.