In terms of short reads I have to piggyback your Vonnegut suggestion with his essential essay collection titled "A Man Without a Country." It's well known but if you haven't read it, as a Vonnegut fan you must. Also I'm not aficionado of the "classics" but you should at one point in your life thumb through some Hemingway. Also not a surprising or discreet suggestion but i think they're both essential collections.
I'm not much of a Vonnegut fan, actually. I tried Slaughterhouse Five, but I couldn't get into it. I don't know why. I really wanted to like it, I just didn't. I like Harrison Bergeron due to the dystopian it is. That's also why I liked The Veldt. Not really a dystopia, but Veldt Spoilers kids so spoiled that they killed their parents? A bit dystopian if you ask me. I also loved 1984 and Little Brother. If you look at some of my stories, namely A man thinks he is the last human on earth, and finds out he's not and The world after pollution, it has been deemed as toxic and one man wants out, they are both dystopians. I don't know why I like dystopian societies so much, I just find them so interesting. If you have any dystopian books, I would love to hear about them. I'm sure there are plenty I haven't read yet.
Slaughterhouse was my least favorite, I recommend giving Vonnegut another try if you ever have the urge. Maybe his satire and style of writing isn't your taste, he doesn't exactly pander to the widest audience. I'll absolutely read your stories and get back to you when I have the chance, but with a few exceptions like 1984, I'm actually not one for sci-fi and the dystopian genre. I guess we gotta venture into each other's playing fields then!
I think Slaughterhouse Five is great, but I am very glad it was not my introduction to Vonnegut. Instead I was introduced with Player Piano I would suggest that or "Cats Cradle" if you give Vonnegut a second try.