The big band jazz is mostly from the 1930s - 1940s. Some of my favorites are Benny Goodman, Artie Shaw, and Glenn Miller.

Duke Ellington and Count Basie wrote some great big band tunes in the 1930s. Thad Jones is also one of my favourites, but his music is a bit later in the 20th century

Dizzy Gillespie is really a must hear for any jazz nut. He has some crazily complex melodies.

I would highly recommend Glen Miller, probably one of my favorite big band jazz artists.