What does the reader learn about Caesar's, Cassius's, and Brutus's characters throughout act 1? Give your points with quotes.

From his initial appearance, we learn that Caesar is a superstitious man. He wants his infertile wife, Calpurnia, to be touched by Antony, since the belief is that the touch of a ceremonial figure can cure sterility. However, this seems to be a circumstantial belief for Caesar, since he ignores the warning of the soothsayer to "beware the ides of March." He dismisses the soothsayer with a curt judgement: "He is a dreamer; let us leave him:—pass" (1.2.24). He also speaks in the third person (using the "royal we"), showing that he has a high view of himself.
Cassius's first major dialogue shows his interest in Brutus and his crafty ways. He comes off like someone looking for an opening, trying to find the best way to win Brutus to his case. He is also observant. We understand this from when he says to Brutus,

And, since you know you cannot see yourselfSo well as by reflection, I, your glass,Will modestly discover you to yourself.(1.2.67–69)

Cassius intends to use his observances to his own advantage, and he is successful when he convinces Brutus that the right thing to do is kill Caesar. He is a sneaky and keen-brained fellow.
Brutus's troubled character reveals itself quickly as well. When Cassius observes that Brutus hasn't seemed himself lately, Brutus says,

Be not deceived; if I have veil'd my look,I turn the trouble of my countenanceMerely upon myself. (1.2.37–39)

This shows that he is a man with high standards in all things, but mostly for himself. He is a noble figure, and that is the biggest thing the audience takes away about him in this initial appearance.