I highly recommend memorizing passages from any play you are enjoying. For some plays there are a wealth of passages worth remembering and for others you have to dig deep but I think it is worth the effort.
Harold Bloom has written quite a bit about the need to memorize Shakespeare. Lea Carpenter broke this down in a 2011 article.
"We are “influenced” by what we remember. Bloom, celebrated and vilified in equal measure, still never wavers from this central theme, and implicit in it is a larger one: literature matters. “Shakespeare is God,” Bloom has said, and one interpretation of that is that one author more than any other has colored the way we see the world. We should show respect.
“Possession by memory is what I try and teach my students,” Bloom tells Tanenhaus. Any actor who has played Hamlet or Juliet knows this is true. Memorization (a comparatively easy task with iambic pentameter) is an excellent conduit to empathy for the poet, a far more powerful and relevant conduit than, say, knowing about his or her sex life."
Over the years I have come to prioritize the memorizing of Shakespeare over other poetry just because it is rare to find better meter or meaning. Five of our last six poems have come from Shakespeare. When in doubt pick Shakespeare.
One thing I failed to mention in the earlier memory sections is how we go about memorizing poems. For the most part we just read the poem every day and gradually begin to leave out words. About 2 weeks into this I will assign the poem as a narration and then in a couple of weeks more I will do it again. When I think we almost have it I will set an end date and push to leave out more and more. Finally when we have learned the poem I put in on my review list every week for a while.
To get you started here are few passages to memorize (and here is another post I wrote) from The Bard:
1. The Quality of Mercy (Portia in The Merchant of Venice)
2. Hamlet's To Be or Not to Be Soliloquy (This is quite delightful with its odd words)
3. Marc Antony's Friends, Romans, Countryman (Julius Caesar)
4. All the World's a Stage (As You Like It)
5. St. Crispin's Day Speech (Henry V, every boy should know this)
7. How Sweet the Moonlight... (Merchant of Venice)
8. The Man that has no music...(Merchant of Venice)
9. Fear No More (Cymbeline)
10. When Icicles Hang (Love's Labour's Lost)
11. Sonnet 29
12. Sonnet 18
13. Tomorrow and Tomorrow and Tomorrow (Macbeth)
14. Thy husband is thy lord, thy keeper..(Kate's speech from The Taming of the Shrew)
15. But soft what light through yonder window breaks (Romeo and Juliet)
16. The lunatic, the lover and the poet ( A Midsummer's Night's Dream)
17. Then you must speak of one who loved not wisely (Othello's death speech)
18. Wolsey's Farewell to his Greatness (Henry V)
19. Give thy thoughts no tongue... (Polonius's famous advice to Laertes, Hamlet)
20. This royal throne of kings, this sceptered Isle.. (Richard II
Suggestion of the Day for Morning Time Memory:
From Richard II by William Shakespeare
This royal throne of kings, this scepter'd isle,
This earth of majesty, this seat of Mars,
This other Eden, demi-paradise,
This fortress built by Nature for herself
Against infection and the hand of war,
This happy breed of men, this little world,
This precious stone set in the silver sea,
Which serves it in the office of a wall,
Or as a moat defensive to a house,
Against the envy of less happier lands,
This blessed plot, this earth, this realm, this England
From Richard II, Act 2 (Shakespeare)