VIOLA: I would stay asleep my whole life if I could dream myself into a company of players.
Viola has a huge obstacle to becoming an actor, and it isn't her acting ability: it's that she is a woman. At the time, on all the stages in the world (or at least in London) all the players are merely men.
VIOLA: Stage love will never be true love while the law of the land has our heroines played by pipsqueak boys in petticoats!
Viola makes a good point here. Can men playing women accurately portray the emotion of love? Maybe the reason no play has shown true love (cisheterosexual love at any rate)is because all the actors are men. Perhaps Romeo and Juliet isn't all that great—it's the fact that a real woman is on stage that makes it so powerful.
VIOLA: You will not tell. As you love me and as I love you, you will bind my breast and buy me a boy's wig!
Here Viola orders her Nurse to disguise her as a man. We're not sure how anyone falls for it before she wears her fake facial hair, but many of the young men who play woman have feminine features anyway.
WILL: Your voice, have it dropped?
SAM: No, no, a touch of cold only.
The men playing women don't have it easy in the theatre. Their careers have a short lifespan. This line foreshadows Sam's voice change, and the end of his ability to play female roles on stage.
VIOLA: I will do my duty, my lord.
Not all the gender issues in this film take place on stage. This one takes place off. As a woman, Viola is not only banned from acting, but she is bound to the whims of her father and Lord Wessex. She is less a person and more like a piece of property to be traded.
VIOLA: I do not know how to undress a man.
WILL: It is strange to me, too.
This is a funny little scene after "Thomas Kent" reveals his true identity as Viola. We don't often see this part of a gender bending comedy, and it adds a bit of levity to a romantic situation.
WILL: Her chaperone. My lady's country cousin. My, but you be a handsome gallant, just as she said! You may call me Miss Wilhelmina!
At this point in the film, Will does his best Mrs. Doubtfire impersonation. William Shakespeare plays, and this movie, have a lot of gender bending, so it's only natural for Will, too, to get in on the act and dress as a nurse for comic effect.
QUEEN: I know something of a woman in a man's profession. Yes, by God, I do know about that.
This is a bit of a throwaway line for some girl power from the Queen, since she isn't that major of a character, but it illustrates why the Queen sympathizes with Viola. She sees Viola as a pioneer in a sea of testosterone, just like herself.