Brian, I can easily see how Brubeck is a smooth transition from classical music to jazz. You can easily recognize a link between someone like Brubeck and chamber music. Let me give you a few suggestions that might work for you:
Benny Goodman was a classically trained clarinetist to such an extent that several composers wrote pieces for him. His small ensembles are his finest works... especially the trios.
This is Brubeck's masterpiece. Every selection was intentionally structured around a time signature removed from 4/4 common time.
MJQ is a quartet and here they build upon or allude to classical music.
If you are exploring jazz you eventually have to get around to Duke Ellington. This selection is as fine as any to start with. But this set (3 discs) is quite likely the finest collection of Ellington's earlier work at his peak:
Ellington and Miles Davis are my two favorite jazz artists... and undoubtedly two of the finest. Coming from a classical background I would probably avoid Louis Armstrong for the present... although he is another towering figure.
Miles Kind of Blue is frequently cited as the single greatest jazz LP ever. I wouldn't disagree:
The record was perhaps the first work of popular music to employ modal structures. But you might find Sketches of Spain... which re-interprets Spanish Classical and Folk music through the trumpet to be even more immediately accessible.
Coming from Brubeck you should check out Bill Evans (who also performed on Kind of Blue). You can't go wrong with a good many of Evans' recordings. I particularly like Waltz for Debby:
Miles can be wild and Ellington and Goodman both swing... but all the discs here showcase the bluesier more "mellow" side of jazz. The same is true of Paul Desmond, Brubeck's longtime sax player.
Take Ten may be his best LP... but I also recomend this:
Thelonious Monk was one of the most innovative of all jazz artists... frequently going against all your expectations. This may be the best starting place with him:
As with any body of music or composer that is new to you jazz really demands you take your time and really listen... often repeatedly. There are very few musical forms or styles I have not been able to appreciate over time.