Source, in full
As quoted in "This Morning" by Shirley Povich, in The Washington Post (Thursday, March 17, 1949), p. 2S; and again on March 20—as "Dickey's teaching me all his experience"—by Arthur Daley in The New York Times; and on March 29—as "Dickey taught me all his experience"—by Arch Ward in The Chicago Tribune.
Notes from Wikiquote
The most distinctive element of this quote as it has come to be universally remembered—i.e. "learning me" rather than "teaching..."—appears to be the invention of Arthur Daley, who, more than a year later (5/22/50 NY Times), reprises Berra's 1949 statement as "Dickey's learning me his experiences."