So I want to start reading the books that won these awards from then to now, but which list should I choose? Should I choose what SFWA, a notoriously stale organization, likes? Or should I choose what the fans, who actually determine if an author continues as an author?
Also, which awards do you value more? For me, it's hands down the Nebulas.
Eh? The two parts of your post seem to contradict each other. The Nebulas are awarded by the SFWA -- hence by what you call "a notoriously stale organization"; while the Hugos are awarded by the readers (or at least a certain number of them).
There are advantages and drawbacks on both counts, but I'm not sure you really need to make a sacrifice either way, as a rather large number are the same in both lists:
Past Winners of SFWA(R) Nebula Awards(R)
The Hugo Award (By Year)
There are some sterling things there, and also some of considerably less merit. (I find Panshin's
Rite of Passage almost unreadable, I must admit.)
However, if you're intending to read both the novels and the shorter fiction, the Hugo winners are a bit easier to track down (overall), as all the short fiction up through the late 1990s or early 2000s has been collected in a smaller number of volumes; while the Nebula winners are more spread out. (And, if you're going to do the Nebulas, I'd suggest getting -- if you don't already have -- the
Science Fiction Hall of Fame, volumes One and IIA and IIB, which are rather similar, in that they represent what the writers picked as the best in the field from Wells'
The Time Machine until the issuing of the first Nebula awards....)
Either way, you've got a lot of good reading in there; enjoy!