I've been a Deadhead for about four decades now. It was a cassette of American Beauty that turned me on to Jerry and the gang. I know that they're not everybody's cup of tea, and that's alright, but I do want to put my own life as a counter-example to those folks who contend you've got to be chemically altered somehow to enjoy the music of the Dead. I'm about as sober as they come (no moral judging, it's just my nature) and I totally dig their country, bluesy, poetic jams. I think, ironically, it's because the Dead are ultimately just so damn life affirming.
Last week on Amazon I stumbled onto this fine Kindle book called The Grateful Dead and Beyond: A Grateful Dead Primer. I didn't need priming, but the book was free and it turns out it's a good little history of the Dead together with a nice discography. Written by Dennis McNally, the Dead's longtime publicist, there are some good stories about the circumstances behind the band, their songs and the legendary tours. I learned stuff I didn't know, learned more about things I'd heard about and generally enjoyed every page.
If you aren't a Deadhead, don't start with this book. Start with their music and you can start pretty much anywhere. Tons of their stuff is publicly available (the Dead are big on that, another thing I love about them). I'll include a few links to some fun Youtube videos and what I think are their best albums.
Keep on truckin'.