Genre is how you classify the written word, generally by the way in which it was written. It groups together those works that are inherently similar. As such, these pieces of literature share similar characteristics, follow similar structures and obey certain conventions. For instance, a poem has different rules than a short story, which has different characteristics than a play or novel. You wouldn't judge Edgar Allan Poe's "The Raven" using the same standards as you would judge Tennessee Williams' "A Streetcar Named Desire" because the two are completely different genres of literature, with different purposes.
Theme, on the other hand, is that abstract idea that pulls the piece of literature together. Literary themes run the gamut of human emotion, and explore each facet of the human experience. The dichotomy of good and evil provides the thematic landscape in works like Harper Lee's "To Kill a Mockingbird," whereas "Romeo and Juliet" explores the powerful, driving force of love. Themes connect the reader with each piece of literature on a more-emotional level by making the message more universal and accessible.
When studying genre especially, you will base part of your criticism on how well the author has fulfilled certain expectations. Poetry can range anywhere from more-liberal free verse to restrictive haiku, which has three lines containing five, seven and five syllables, respectively. A play or screenplay will be expected to conform to a more-rigid structure. A first act will introduce character and conflict, whereas a third act will resolve what was built upon in act two.
Writers often use symbolism that will underscore the underlying theme of their particular work. Because of its wonderful ambiguity, this can fall under more than one different school of literary criticism. In Emily Dickinson's poem "280," she used the symbolism of death to explore themes of repression for a feminist study. In "The Great Gatsby," F. Scott Fitzgerald used the fall of 1920s America to explore the much-larger theme of the corrupting influence of excess, which you can explore through deconstructionist literary criticism.