How do you learn in the first scene that something is wrong in Denmark in Act 1 of Hamlet?
There are actually two distinct problems in Denmark that we learn a little about in scene 1. Typical of Shakespeare, he establishes the mood and the initial conflicts in the first scene, and then has the details of the story unfold in the subsequent scenes in Act 1.
The first problem that we see is that there has been a ghost wandering the castle grounds for a couple of nights. The guards know what they have seen, but have asked Horatio, a school friend of Hamlet's to come out and verify it. Horatio recognizes the appearance of the ghost to resemble the recently deceased King Hamlet. He tries to communicate with it, but the ghost merely stalks away. Horatio tells us the appearance of a ghost can have no good cause. Ghosts come back to reveal a secret; to finish unfinished business; or to reveal treasures. Even worse, the ghost could be a devil in the disguise of a loved one and out to damn the soul of a living person by enticing them to commit some foul act. The arrival of a ghost is never really a good thing according the Renaissance understanding of ghosts.
The second thing that is wrong in Denmark is that Prince Fortinbras of Norway has "sharked up a list of lawless resolutes" -- mercenary soldiers -- to attack Denmark in the hopes of his regaining lands that his father lost in a battle with King Hamlet many years before. This external threat to Denmark is the reason, Horatio reports, for the increase in military preparations going on seven days a week.
It is through the minor character of Horatio that we learn about these over-arching conflicts for Denmark. By the end of Act 1 we know a lot more what the ghost represents, and the play then progresses through rising action to the climax and resolution of each of these conflicts.