In the first two acts that I have read, you can see that a lot begins to unfold. Hamlet, still grieving for his father, is about to return to his schooling, yet stays as requested by his uncle and mother. He soon learns that there is a ghost in the shape of his father Old Hamlet who appears at nights, so he decides to investigate. He speaks to the ghosts and learns that his father is cursed to purgatory because he was actually murdered before he was able to confess his sins. Old hamlet asks his son to avenge his murder by killing the murderer who is actually his uncle, Claudius. All while this is happening, Laertes, son of Polonius, is about to venture to France. Yet before he leaves he tells his sister Ophelia to ignore the lies Hamlet speaks to her and wait till he becomes king before she agrees to marry him. .
Claudius learns that Fortinbras has gathered an army against Denmark and decides to send his uncle a letter asking him to speak to Fortinbras. Old Norway, the uncle of Fortinbras, was unaware that his nephew was raising an army against Denmark. Claudius learns that Old Norway summoned Fortinbras to meet him as soon as he heard about his nephew's plans. Fortinbras complied with the summons and was forced to vow to never attack Denmark ever again and was allowed to attack Poland instead. Old Norway also wrote a letter to Claudius asking him to allow Fortinbras a safe passage through Denmark on the way Poland.
Polonius convinced that Hamlet has gone mad brings a letter from Hamlet to Ophelia in which Hamlet professes his love to her to his mother and father. Gertrude then asks Polonius how Ophelia received Hamlet's confessions of love, and Polonius tells them that at his request she ignored Hamlet. Claudius is not completely convinced that this is the full cause of Hamlet's insanity, so he and Polonius decide to put Ophelia into the hall where Hamlet is known to spend hours pacing each day and hide behind a tapestry and see what happens.