I've never been super keen on short stories.  I don't have any particular reason for this to be so, but the medium never clicked with me.  And yet I like science fiction, and classic science fiction was often born of short stories, often published in digest magazines that collected and serialized works.  Lots of famous sci-fi authors had their debut this way, Philip K. Dick being one of the more notable. 

Dick is probably most well known for Do Androids Dream of Electric Sheep? which became 1982's Blade Runner, but there have also been a few films over the years that have been drawn from his short stories, including Minority Report, Paycheck, Screamers, The Adjustment Bureau and Total Recall (1990 version with Ahnold and the upcoming Colin Farrell verison). These stories have been collected many times in various permutations under various titles, with reprints often following the release of yet another movie version.

His work is quite varied.  His early career starts of as pretty straight forward action/sci-fi with twisting plots that often revolve around messing with technology, sort of mad-scientist kind of stuff (just look at the movies I listed aobve), but gradually gets more cerebral and philosophical later in his career.  The short stories are a great entry into his body of work for that reason, and that it's easy to skip the boring ones to move on the more exciting plots.  Not all of it is gold, but with dozens of stories, there's usually something for everyone (or at least every type of sci-fi fan).  As usual, I wouldn't

There are many collections, but I recommend: 

We Can Remember It for You Wholesale
Second Variety
The Short Happy Life of the Brown Oxford
Minority Report

These are a good start, and doesn't even cover his long form fiction which I'm not so quick to recommend.  Maybe I'll get into that some other day.