A lot of key elements of the Bat-mythos weren't introduced until the 1940s. Batman's home was first named as "Gotham City" in Detective Comics #48 in 1940, and before that Batman lived in "Metropolis" or just New York. The Bat-signal didn't appear until Detective Comics #60 in 1942. The Batcave didn't appear until 1948, and prior to that Batman just had a secret hangar for the Batplane and a deserted barn connected to Wayne Manor by a secret passageway. Prior to the appearance of the Bat-signal, Batman covers frequently depicted a giant moon (which was frequently bright yellow) behind Batman. (Source: The Golden Age of Batman by Artabras).
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