If there is one Charlie Chaplin film to recommend, as others have pointed to in the past, City Lights is the one. Though Chaplin played his Tramp character superbly in other movies, like Modern Times and The Gold Rush, City Lights displays the Tramp at his funniest, his bravest, his most romantic, and his most sympathetic. It's tough for filmmakers in recent days to bring the audience so close emotionally with the characters, but it's pulled off.
The film centers on three characters- the Tramp, the quintessential, funny homeless man who blends into the crowd, but gets caught in predicaments. He helps a drunken businessman (Myers, a fine performance in his own right) from suicide, and becomes his on and off again friend (that is, when it suits him and doesn't notice his 'friend's' state). The other person in the Tramp's life is the Blind Flower Girl (Virginia Cherrill, one of the most absorbing, beautiful, and key female performances in silent film), who are quite fond of each other despite the lack of total perception. The emotional centerpiece comes in obtaining rent and eye surgery money, which leads to a (how else can I put it) magical boxing match where it's basically a 180 from the brutality and viscerality of a match in say Raging Bull.
Though there is no dialog, the film achieves a timelessness- it's essentially a tale of two loners who find each other, lose each other, and find each other again (the last scene, widely discussed by critics for decades, is moving if not tear-inducing). And it's never, ever boring- once you get along with the Tramp, you find the little things about him, the reaction shots, the little things he does after the usual big gag (look to the ballroom scene for examples of this, or when he gets a bottle of wine poured down his pants without the other guy noticing). Truth be told, if this film makes you indifferent, never watch Chaplin again. But if you give yourself to the film, you may find it's one of the most charming from the era, or perhaps any era.