“Pressing charges” basically means you will cooperate with a criminal investigation and court case, if it gets that far. You can’t press charges if the police do not have enough evidence to issue a ticket or charge the person with a crime.
You can also sue in civil court to recover monetary damages (as in, costs to replace things, etc.). You do not need as much evidence as a criminal case, but if you don’t have good evidence, you are liable to lose the suit, and may have to pay for the other party’s lawyer and costs.