Free will versus rational determinism
The Underground Man's core argument is that humans can't be reduced to calculable beings who want what's good for them. He insists people will act against their own interest to prove they're free. This theme drives all of Part One and plays out in his actual behavior in Part Two.
Self-consciousness as paralysis
He thinks too much and acts badly as a result. Every social interaction gets analyzed before, during, and after it happens. That constant self-monitoring doesn't make him better. It makes him unable to act naturally or connect with anyone.
Resentment and wounded pride
Almost every action in Part Two comes from humiliation. The officer, Zverkov, his schoolmates, even Liza all become targets for his stored-up resentment. Dostoevsky shows how a life of small social wounds can curdle into something destructive.
The gap between thought and action
The Underground Man can articulate what's wrong with himself and with society. He cannot change either. That gap between knowing and doing is what makes him tragic rather than just unpleasant.
Isolation and the need for recognition
He claims to prefer solitude but spends the entire novella seeking an audience. He writes to an imagined reader he simultaneously mocks. He crashes parties he wasn't wanted at. He wants to be seen and refuses to admit it.