The most important thing Nick learns about Tom by the end of chapter 2 is that he is having an affair.
During the chapter, Nick and Tom travel to pick up Myrtle and they meet by catching the train from different locations to the city. As they travel and arrive, Nick notices how open they are about their relationship. If someone who knew either of them saw, it would be obvious. At the very least, they keep this relationship out of the view of their spouses.
By the absolute end of the chapter, Myrtle has turned from blue collar wife to high society prom dress gal. She has Tom buy her a dog and they all spend the evening together in an apartment Tom apparently keeps for the sake of this affair. At the end of the chapter, she taunts Tom by saying Daisy's name over and over. This obviously shows Tom's disgust with his own wife who happens to be Nick's cousin.
This Chapter features lots of information about Tom and his character as he forces Nick to accompany him to the house of his lover, Myrtle, taunts her husband then takes Myrtle and Nick with him to his appartment that he uses for affairs and has an unplanned party with Myrtle's sister and a couple called the McKees. They drink a lot, and although Nick is disgusted by their exaggerated behaviour, he is compelled by the novelty of observing these shallow people. When Tom gives Myrtle a puppy, she begins to talk about Daisy. Tom tells her to keep quiet, and Myrtle refuses, chanting Daisy's name. Tom then breaks her nose in an act of violence, bringing the party to a sudden halt.
This party scene clearly establishes the hypocrisy of Tom and also his inability to control his emotions, especially his anger. He has no guilt about his affair with Myrtle, but feels he must keep Myrtle in her place and control her. His violent act against Myrtle illustrates his character: he is a physical bully who uses and abuses his social position and physical strength to dominate and manipulate those around him. Note how he taunts Wilson whilst also cuckolding him. He has no guilt about his behaviour and without a doubt is quick to violently assert his authority over Myrtle. Wilson here acts as a foil to Tom - he is an attractive and moral man, but who doesn't possess the money, vitality or position that Tom has.