I am writing an essay about Hamlet's loneliness and depression with a study of his various friendships, what would be a good intro for this topic?
I am writing about how Ophelia, Rosencrantz and Guildenstern, and his mother, and how they all contribute to his sense of isolation.
A good introduction on a topic like this would be to talk about the importance of friendship and the influence of friendship on people in general, and then narrow the focus of the introduction to talking about how Hamlet's friends (for better and for worse) affect his actions in the play. This kind of introduction is called a funnel: broad at the top, narrow at the bottom. Sadly, it is the very people who should make Hamlet feel better that actually make him feel isolated because they all prove so disloyal to him. That needs to be the ultimate point you make; that is your thesis statement.
Rosencrantz and Guildenstern are the strongest examples you have for this thesis because while they come to Hamlet in the pretense of friendship, we know that their ultimate motivation is to serve the king in hopes of a reward from him. They are given specific instructions to find out what is the source of Hamlet's "much too changed" ways and that goal is never far from their thoughts. Hamlet begs them to just tell the truth whether they were sent for of not (by the King and Queen). When they don't immediately tell the truth, Hamlet, in an aside, says "I have an eye of you." He is telling us that he sees the truth of these so-called friends.
He feels betrayed by Ophelia when in the middle of the "get thee to a nunnery" conversation he asks where her father is and she replies "at home." In truth she knows that he is in hiding, observing this whole conversation, and Hamlet seems to know that too as revealed by his next comment: "Then let him be a fool no where than in his own house." (He certainly isn't going to get away with playing these games in Hamlet's house, the castle.)
It might be an interesting component of your essay to write about how Hamlet's only TRUE friend in the whole play is Horatio. You could write about how Hamlet speaks in a way that raises Horatio's status from "servant" to "friend" as seen in his first conversation with Horatio in Act 1. Horatio is NOT a servant, but he is not of noble/equal status to the prince. In a later conversation in Act 3, Hamlet specifically expresses that he admires that Horatio has been able handle what Fortune has thrown at him with a balance of "blood and judgement." In the end, Horatio is the loyal friend who lives on to tell the truth about all that has happened in Denmark that has ultimately brought them to the present disastrous ending.