Kind of a very fun lifetime of work... But you are going to have to put some time into it to mean anything.
One way I learned is to just surf the reviews in Amazon - one artist leads to another etc. Also some very good books on the subject:
Jazz Styles: History and Analysis by MarkGridley gives you pretty much exactly what you are looking for. It breaks Jazz into a series of eras and tells you who the key players were in each era.
The Penguin Guide to Jazz Recording lets you see an artists entire body of work with the editors comments about which of the recordings are the most significant - help you separate the wheat from the chaff as you get started
One other thing - these guys were incredibly prolific and recorded for many decades (not like the rockers 3 albums and out). Most of them recorded constantly. And record producers were fond of all kinds of pairings. So you can pick a seminal figure like a Ben Webster and listen to him play with any number of other greats. This is a nice way to compare and contrast.
Let me offer up a few names to get you going. I'll use the structure from Jazz Styles. You can then browse Amazon - focus on the 5 star recordings and just spend some time reading and listening to samples - its all free.
Up to the early 30s. Sax was not widely used then - this was piano, trumpet and clarinet time coming out of New Orleans Dixieland traditions and Chicago blues. Jelly Roll Morton and Louis Armstrong were the huge names - as was Benny Goodman. Key early sax figures were Bix Beiderbecke (also on trumpet); Sidney Becket and the Hawk, Coleman Hawkins.
There is a lot of great music pre 1940s but I tend to start in the 40s as the recording quality is really much better.
From the 30s to the late 40s was Swing. This was the era of the big bands, and during this time sax emerged as an important solo instrument - Coleman Hawkins was the man. Also mentioned is a guy named Don Byas whose work I don't know. Benny Carter was the guy on alto and Johnny Hodges was part of this era though he came to fame as part of the Ellington bands.
This era also marked the beginning for Billie Holiday and Ella Fitzgerald. Billie later duetted a lot with the legendary sax player, Lester Young. And if you want to hear the absolute pinnacle listen to the albums by Ella and Louis Armstrong.
This brings us to Chapter 7. Simply entitled Duke Ellington. Not only was Duke beyond brilliant as a composer and arranger; his band was a kind of incubator for some of the most famous players. And it was in his band that the art of the big band solo was perfected since the Duke was famous for creating arrangements that took advantage of each players individual style.
Johnny Hodges was the man on alto and he influenced an enormous number of people. My main man on tenor was Ben Webster - who emerged as the sax star in the Ellington bands under the tutelage of Johnny Hodges. For me "the Brute" is the heart and soul of all things saxophone. The other Ellington sax player was Paul Gonsalves whose work I don't know nearly as well.
The other famous and much loved big band was led by Count Basie who was a very different kind of pianist and leader then the Duke. (Basie came out of the stride tradition and was famous for his comping style.) Definitely listen to Duke Ellington Meets Count Basie.
Probably the biggest sax name out of Basie's bands is the aforementioned Lester Young, the Prez. Basie was from Kansas City, and the other big name from KC is Charlie Parker.
The Prez was to influence the cool jazz movement. He is cited as an influence on such diverse sax players as John Coltrane and Stan Getz, Zoot Sims and Dexter Gordon and many others (those are a few of my faves)
Swing gradually evolved into modern jazz called Bop or BeBop (earlyy 40s to mid 50s). Hawkins and Young were seminal influences on the sax. Other key names were alto sax player Charlie Parker, trumpet player Dizzie Gillespie and piano player Thelonius Monk.
Other key names that emerged in this time were Oscar Peterson a brilliant piano player who played with pretty much everyone, sax players Sonny Rollins and Sonny Stitt, my man Stan Getz and bari sax man Gerry Mulligan.
Some of the other sax players influenced by Charlie Parker include: Bud Shank, Cannonball Adderley, Eric Dolphy, Art Pepper and Ornette Coleman.
Next up was the era of Cool Jazz. Note that where swing and the big bands had played what was essentially dance music, no one went to hear cool jazz to dance.
There were two major camps. The West Coast style was epitomized by the Woody Herman and Stan Kenton Big Bands. The influential sax sound was Lester Young. My favorite sax players of this group are Jimmy Giuffre who also played clarinet and Zoot Sims. This was also where alto player Paul Desmond who played with pianist Dave Brubeck came from.
The other school (never called east coast as far as I know) was Hard Bop. This is a rawer, darker approach with more complex melodies then West Coast.
This is Cannonball Adderley, Sonny Rollins, John Coltrane, the under appreciated Stanley Turrentine and Joe Henderson plus Wayne Shorter who went on to play with Herbie Hancock and Yusef Lateef.
Miles Davis gets his own chapter as he was both a trumpet virtuoso and a brilliant bandleader (check out Sketches of Spain and Kind of Blue). For me Miles and Coltrane go together.
Coltrane gets his own chapter too. While he is not my favorite sax player, like Miles his work is very influential. Giant Steps and A Love Supreme will get you started.
During the 60s and 70s was a period known as Avant-Garde or "Free" jazz. Some weird stuff got done - paralleling the exploratory work being done by the Beatles and other acid loving rockers.
Alto player Ornette Coleman emerges as one of the influential voices in this movement - check out his album Free Jazz. He recorded with a lot of guys who are among my favorites including bassist Charlie Haden and Bill Evans bassist Scott LaFaro.
Charles Mingus is another emerging light in this genre. And a sax player named Albert Ayler whose work I am not familiar with.
You can sum up this whole movement by saying that it never caught on - for most people, including me its simply not very accessible.
Next chapter is Bill Evans (one of my all time faves on piano) who influenced Herbie Hancock (brilliant keyboardist and composer with a huge discography), Chick Corea (ditto) and Keith Jarrett (another pianist).
There really was no sax man associated with this except for Wayne Shorter with Herbie Hancock; and Dewey Redman with Keith Jarrets quartet. (BTW Wayne plays the sax on Steely Dans landmark Aja which is pretty darn close to jazz IMHO)
Next is Jazz Rock Fusion with the very pretty sax of Grover Washington (put on Winelight and get cozy). Other names include Ronnie Laws and on alto David Sanborn and Kenny G. The soprano and alto sax were much favored for their ability to ride above the rest of the band.
A lot of the names we still reference today were part of this era: John McLaughlin, Pat Metheny, Larry Coryell (all guitarists)
The music goes on into the more contemporary sax players including Joshua Redman, the very brilliant Branford Marsalis (Eternal) and the very intense Charles Lloyd (The Water Is Wide).
But for the most part the golden era of jazz sax, the sounds of the Brute, the Hawk and the Prez are behind us.
Go buy the books, go to Amazon, go to Wikipedia, listen to your local jazz station or one on the Internet and let us know what you find and what you love.
This has the potential to be an enormous and wonderful thread.