EARTH ABIDES—George Stewart
ALAS, BABYLON—Pat Frank
These two books reveal the current crop of zombie apocalypse books as the utter garbage they are. After you’ve read those, you can move on to what I consider the great novel of post-apocalypse America...
A CANTICLE FOR LEIBOVITZ—Walter M. Miller, Jr.
I don’t want to muddy the waters with too many suggestions. People should read these ones. The Miller influenced one of my first published short stories, but I think one can objectively note that it’s considered a classic.
I have not read Earth Abides but will look for it. I have read the other two and recommend them, especially Canticle for Leibovitz.
I also recommend
Larry Niven Lucifer’s Hammer
and
Steven King The Stand