Themes are the points that the author is trying to make with the reader/audience. Some of them will be obvious to you and others you will have to think on for a bit. Most of the time, you can figure out the main themes of a piece by saying, "_______________(title of the work) is a story about ________________________(first thing that comes to your mind)." It almost always deals with universal ideas like love, greed, ambition, hate, pride, good vs. evil, etc.
For Hamlet, there are also many themes. Ambition, reality vs. illusion, illness, madness, revenge, murder, love, friendship, honor, and family relationships just to name a few. Check out the link below to help you find more and to get a more analytical look at how these themes play out through the work.
You've come to the exact right place. Enotes contains in-depth analysis on themes of thousands of different stories, plays and poems. I have provided a link below that will take you right to a discussion of the various themes that are contained within "Hamlet".
In a nutshell though, there are several possible themes. An obvious one is revenge; Hamlet's father comes to him and tells him to enact revenge on his murderer, and Hamlet spends the entire play attempting to do so, do the detriment of everyone around him, and himself. One has to ask if that revenge was worth it, and what Shakespeare was trying to say about revenge as everyone in the play is destroyed by Hamlet's quest for it. Another possible theme is that of madness. All of Hamlet's friends and family are wondering if he is indeed mad; even the audience is put off by his "antic disposition" and wonder themselves. It seems more probable that he is pretending in order to not arouse supsicions as he investigates his father's murder, but it is interesting that he would choose the guise of madness to do it under. One last theme I'll mention is that of death; Hamlet's father dies, life is miserable to him, and he spends a lot of his time moping about wondering if he just shouldn't die too. He ponders why we don't in his famed "To be or not to be" monologue, and spends a great deal of time mulling it over as he holds the skull of a departed jester, Yorick. So, death is another pervasive theme in Hamlet.
I hope that helps to get you started; here's the link: