I'm a bit late to the party and much of what I would say has been said. In particular Gary above has made the essential point that they were never just one band and went through distinctive phases. The phrase from the Merry Pranksters, going back to the earliest days of the Dead in San Francisco and experimentation with psychedelic drugs, multi-media shows and wild music, was that you were either on the bus or off the bus. I would say that if you don't get the Dead, then leave it. To test the water I would say try three records from early in their career:
1. Europe 72 (this is a live album with some post recording overdubs). For many, that European tour was one of the very highest points of the Dead's career and the album gives a good sense of the variety of styles in which they were interested then and includes some more extended jamming as well as some great songs. If you don't think Jerry Garcia is a great emotional singer after listening to this, you probably won't ever get on the bus.
2. American Beauty or Working Man's Dead. These were the Dead's Americana albums - a delightful semi-acoustic mix of blues, folk, country and rock when they first really concentrated on singing in the studio. These were their best sellers so should be the most accessible version of the Dead. The Dead's version of psychedelia never followed the twee English version of fairies and hobgoblins (a gross generalisation, I admit), but is more closely related to the folk tradition in subject matter. See for example Box of Rain on AB which sprang from Phil Lesh's (bass player) father's death and Black Peter (from WD), which is about the thoughts of a dying man.
3. Live/Dead. You will either agree that this is one of the greatest live rock albums ever recorded or get bored with the 'noodling'. In fact the album again demonstrates the versatility of their playing, from the intricate intertwining of the different instrumental voices on Dark Star, to the complex rhythms of the Eleven, to Pig Pen's channelling of raunchy blues and soul on Turn on Your Lovelight.
I would say that if nothing here excites you, the Dead are not for you. If it does, then there are many avenues to explore as indicated above.
For me, if I had to choose just one album it would be their second, Anthem of the Sun, an astonishing blend of live and studio recordings that marks the exploratory energy of a band that has no limits on its ambitions - I really think there is no other rock album anything like it.