Select one or two famous monologues such as Hamlet's "To be or not to be" and one of the many famous monologues from classics like "A Streetcar Named Desire" and then read them in the context of the whole play instead of just reading the monologue on its own. This will help you explore the characters and themes of the drama through the monologue, giving you a better idea of the monologue's dramatic power.
Check out famous monologues from film since these monologues are often more accessible and familiar. Start out by searching for famous monologues in a monologue archive such as Colin's Monologue Archive. Then, read through the script of the film, including the monologue. Finally, explore the monologue one step further by watching the film and paying attention to the actor's performance of it.
Search the Monologue Database for a broad list of monologues from contemporary and classic plays. Select monologues by topic or author and then read a few, one after the other, to compare the effect they have on you. If you find one that's particularly moving, check out the entire play to see how the monologue is situated in the larger dramatic context.
Watch different performances of the same monologue, whether it's a film or theatrical monologue, to explore the flexibility of the text and how one performance can exaggerate the monologue while another can downplay its importance. Look for film remakes and different theater productions of the same play to do this.