Whatever you go with, always read Shakespeare out loud. That is how you can feel the contribution to English through the prosody of his writing.
It's hard to really recommend what you should read first, since I don't know you. But when I was in high school we read Romeo and Juliet first. Hamlet is pretty great for very young people too, as Hamlet is like 17.
I found Richard III more interesting than Macbeth, personally. Macbeth has more spooky stuff and the theme of guilt whereas Richard III has a lot more to do with interpersonal drama. I'm just going to free-flow ramble about the different ways these particular plays could be connected (and so a sort-of suggestion of how to read them). Always read Shakespeare out loud. You could easily get bored or confused with the text otherwise, and like I said, you start reading out loud like an actor, you'll understand why he's still so celebrated today.
Othello - Hamlet - Macbeth - Richard III : a cycle on tragedy, villainy, choice, hate. Read these in any order; my order has to do with a weird hero vs villain virtue math that I'm still debating with myself over.
Romeo and Juliet - Antony and Cleopatra : there are enough parallels to enjoy these two together
Julius Caesar - Antony and Cleopatra : IMO Marc Antony may have been one of Shakespeare's favorite heroes from ancient times and you see a lot of him in these.
Much Ado About Nothing - A Midsummer Night's Dream : comedies more to do with frivolity, antics, and folly - as opposed to idiocy or lunacy (Shrew)
The Taming of the Shrew : I read this recently, this is a rather standalone play as its comedic literary conventions are (and were historically) rather rough if you forget the induction. I would read the Wikipedia page about this ("Analysis and Criticism" - "Themes") afterword to complement it.
I would pick up a copy of Twelfth Night or even Two Gentlemen of Verona before you read The Taming of the Shrew to get a better idea of Shakespeare's "typical" comedies...Shrew does not have a true "center of good" character IMO, which makes it one of the more challenging of Shakespeare's plays to reconcile with (especially as a comedy): everyone sucks and you are left wondering wtf sort of world is this. The framing technique Shakespeare uses at the start of the play is essential to understanding "the joke" but for a lot of people historically, it is not enough and the play can leave the audience a bit uncomfortable. The movie 10 Things I Hate About You is an adaptation that really brings together the story and modernizes it in a way that is hyper-palatable and lovable.
Thanks! I have never considered reading aloud before. Will do that.
I love how you batch certain plays together so that I will focus on the common themes and have a more consistent reading experience haha