There really is only one thing that Hamlet makes Horatio swear, in Act I, but Hamlet makes Horatio swear it at two different points.
The first thing that Hamlet makes Horatio swear is that he will not tell anyone about the fact that he has seen the ghost of Hamlet's father. Hamlet asks Horatio to swear to this in Act I, Scene 2. Here is a quote that shows this:
I pray you all,
If you have hitherto conceal'd this sight,(265)
Let it be tenable in your silence still;
Later on, Hamlet actually sees the ghost and talks to him. After he has done this, Horatio and the others come in and talk to him. He again makes them swear not to talk about what has happened. Here is the quote to show this. It is in Act I, Scene 5.
Come hither, gentlemen,
And lay your hands again upon my sword.
Never to speak of this that you have heard:
Swear by my sword.