Well the answer isn't entirely easy. Perhaps the most accessible are Workingman's Dead and American Beauty, in that these albums feature songs and not a lot of improvisational jamming. However, if you are more interested in the jazz side, then those albums might not be the best place to start. Another nice place to start might be the released live albums such as 1971's The Grateful Dead and 1972's Europe 72. Yet another really interesting place to start would be Wake Of The Flood, or Mars Hotel, or Blues For Allah. These albums have some very nice writing on them. However, for just starting out, Wake Of The Flood is probably a little less accessable than Mars Hotel or Blues For Allah. It's difficult narrowing down from there without knowing what kinda stuff appeals to you. If I had to pick one, Mars Hotel might be a good starting point, although I'm sure there will be many folks around here who will disagree with that choice ! I recently played Help On The Way/Slipknot from Blues For Allah and Estimated Prophet from Terrapin Station for a friend of mine who was unfamiliar with the Dead and he becase an instant convert!
Having said all that, a different approach would be to download some of the stuff that's free and legally available from the huge number of taped shows. There are lots of place on the net where these downloads can be found. So if you downloaded a really good live version of Help On The Way/Slipknot and you really liked it, then you could purchase the studio version (the sound quality of which is great). However, you would need to keep in mind that the Dead were not highly consistent in the quality of their live playing. Still, there is a lot of really nice stuff out there.
One place you probably would not want to start out at is any of the first three albums. As an introduction, they might be just a little bit strange!