I would recommend "Serenity" (the two-hour pilot), "Out of Gas", "Ariel", and "Shindig" as the best ones for introducing what the universe is essentially like.
The pilot sets everything up; "Out of Gas" gets you the backstory, some cute science, clues you in on what other people in the 'verse are like, and is a great character study; "Ariel" shows you the Central Planets, is a great heist story, establishes the freakier aspects of the Alliance's operations, and showcases several key personalities - Mal, Simon, Jayne; "Shindig" has good character stuff, shows you what the world looks like from several different levels of society, and establishes several aspects of the 'verse that are worth knowing about - slavery, cattle being incredibly valuable property, and the role of companions.
"Objects in Space" is one of my favourite episodes, but I firmly believe its impact is massively diminished if you haven't seen the entire series. Similarly, "War Stories" and "The Message" have very important character development, but mean less if you're not already grounded in the show's universe and invested in the characters. "Heart of Gold" is a really good story focusing on one aspect of the 'verse and has vital character development, but both elements make it well-suited to its place near the end of the run.