Metallica is a band that carries so much personal, cultural and historical baggage that it can be very intimidating trying to decide where to dip your toes.
Most people will tell you to listen to their first four albums or less whilst some will tell you to check out their latter albums. Some will tell you to ignore them completely citing "real" Metal bands to check out instead.
Personally, I think you would be best served checking out a "best of" album that covers their entire career.
If that proves difficult or if you want to start with a proper LP then yes, I suggest starting with one of their first four albums. Master of Puppets is probably the best place to start as it offers a nice mix of fast and slow songs, heavy and light. From there I suggest listening to the other three eighties albums: Ride the Lightning, ...And Justice For All, Kill 'Em All.
That concludes the Bay area Thrash portion of Metallica's career. After that they started to move away from mid-fast paced Thrash and move towards "Classic" Meta with their 1991 self-titled album Metallica - or The Black Album as many like to call it due to the album art being 95% black, a reference to the Beatles White Album.
Many argue that The Black Album was when they "sold-out" and abandoned their core fanbase. Some cite radio friendly songs like Enter Sandman and Unforgiven as evidence. Others place the blame directly on them them cutting their trademark long hair. Either way, The Black Album was a shift in Metallica's musical career but despite what some of critics say the album is an excellent one - just not a Thrash album.
After a five year long touring campaign, Metallica returned with their most controversial album, Load. labeled by some wags in the press as "Metallica does Country", Load is a brooding, introspective album that shifts the focus away from vigorous guitar licks to heavy, rumbling chords backed by some very dark lyrics, easily the best of James Hetfield's career. This was also the album that saw Hetfield cement his reputation as the best singer in not only Metal but Hard Rock as well. Long gone are the adolescent barkings of his youth, replaced by a deep, mellifluous tone that is as sorrowful as it is beautiful.
After Load Metallica released Re-Load, basically an album that featured all the songs leftover from the Load sessions. Whilst flawed in some sections it's still a great album, adding a bit of colour to the black-as-coal Load sessions.
From here Metallica released a few non-studio albums, including Garage Inc. (a collection of cover songs, new and old) and Symphony and Metallica (or S & M).
S & M, for all intentenst and purposes could be argued as Metallica's best album, as it not only collects their best songs but complements them with the San Fransisco symphony orchestra under the practiced hand of composer Michael Kamen. Not only does this seemingly bizarre juxtoposition of Classical and Metal work but it in some cases improves upon the original songs. See Master of Puppets as a perfect example. That the album contains two new songs created expressly for the concert rank amongst the bands best and you have a must-have album.
Finally, we have St. Anger. Metallica - after years of condemnation by jilted fans - decided to go "back to their roots" to appease the old-school crowd. The end result, well, it wasn't pretty. If you want to know what went wrong then I suggest you watch the Some Kind of Monster documentary.
The point of this ramble is to give you a sense of perspective and to not discourage you from getting any of Metallica's albums. You can see for yourself how wildly opinions differ on what is a 'good' album so I say to you: Don't limit yourself. By all means get their first four albums but don't be discouraged in seeking out their later ones - yes, even St. Anger. You just don't know whether you'll warm to it or not.
I hope this has helped.
Enjoy!