“Crush Update”
Although the verb crush can mean to squash, have a crush on someone means to be romantically interested in them. The noun crush can name the feeling or the person. Crush Update means the latest news about that romantic interest.
“Quick test”
Literally a "short test," here it means "a little quiz": a light way to check what someone knows. Tina already showed in A2-01 that she remembers Frank likes black coffee. Frank has not forgotten that; as Tina lists one fact after another about the guy she likes, he quizzes her on his own preferences.
“How do I take my coffee?”
Here take means to have a drink a certain way. It asks how someone likes their coffee, for example with sugar or milk. Asking someone How do you take your coffee? is a standard question, answered with something like Black, please.
“Please!”
Here Please! dismisses a suggestion rather than making a polite request. Tina means “Come on—of course I know how you like your coffee!”
“I know it by heart.”
By heart means from memory. Know something by heart means to have it memorized. Tina insists she knows Frank's coffee preferences perfectly well.
“Rude!”
Rude! normally calls something impolite, but friends also use it as a playful protest. Tina objects to being teased about having a new crush every few months. Her quick admission that it is partly true shows she is not seriously angry.
“I've gotta run.”
I've gotta run. is a standard way to cut a conversation short because you have to leave: "I have to go." Gotta represents a casual pronunciation of got to, and have got to means "must." Here, before leaving for class, Tina invites Frank to the game: Will you come with me on Saturday?
“to watch you watch him”
Watch someone do something means to see someone doing it. Here you is Tina and him is the basketball player. Frank says he will go to watch Tina watching him. The punchline is a friend's teasing: besides the game, he is looking forward to seeing Tina cheer, completely absorbed in her crush.