I posted something similar on the Hugo Recommendations LiveJournal. Here it is, updated, with better info and sources on where you can obtain the comics legally.
Captain America Comics #1
Written by Joe Simon and Jack Kirby
Illustrated by Jack Kirby
Timely Publications (Timely Comics?)
It's the one with Cap punching Hitler on the cover. It says March 1941 but the comic came out in December of 1940. One of the finest of the Golden Age, and quite possibly one of the most important. It publicly denounced the Nazis BEFORE the US entered WWII, and while there was still some strong pro-Nazi sentiment in the states. Simon and Kirby's creation, for all its Golden Age goofiness, still stands out as a work of bravery.
Available on Comixology
The Spectre!/The Spectre Strikes! (More Fun Comics #52/#53)
Written by Jerry Siegel
Illustrated by Bernard Baily
National Allied Publications
A shockingly mature story for the 1940's that holds up pretty damn well. Joe Corrigan being denied heaven after his death and being forced to eradicate all evil is an excellent backstory, and makes all his actions understandable.
The issues are extremely valuable, so if you want to obtain them legally, the only place I could find was the reprints in this absurdly-expensive volume, which is a shame. Just take my word for this one?
Written by Whitney Ellsworth, Bill Finger, Paul Gustavson, Bob Kane, Guy Monroe, and George Shute
Illustrated by Paul Gustavson, Bob Kane, Sheldon Moldoff, George Papp, Raymond Perry, and Jerry Robinson
National Allied Publications
There's a reason historians call this the best single issue of the Golden Age...well, two reasons, actually: the Prince Clown of Darkness and the Princess of Plunder. Not only was this Batman's first solo comic, it also had the first appearances of The Joker and Catwoman, in stories that perfectly demonstrate why they've had lasting appeal. There's also a pretty good Hugo Strange story here.
Available at Comixology
Introducing Captain Marvel! (Whiz Comics #2)
Written by Bill Parker
Illustrated by C.C. Beck
The first appearance of Billy Batson and his older Captain Marvel alter-ego. It's an engaging, simple story executed really well, with underpinnings of mysticism that only reveal themselves upon re-reading. It just works.
In the public domain; download here
The Origin of the Spirit
Written by Will Eisner
Illustrated by Will Eisner and Joe Kubert
Register and Tribune Syndicate
Why would I forget Eisner? This is probably the one that's aged the best, with the art looking strikingly modern, even well into the 21st century. While not at the height of its post-war years, The Spirit still came swinging from day one, with its chronicling of Denny Colt's rebirth as the titular character that gradually became a superhero. Extremely influential to the medium. (Also, Ebony White's only in it for one panel. So there's that.)