Well, you have avant-garde composers in so called art music.

Iannis Xenakis - Jonchaies He used different mathematical models for composing, making a kind of architectural music. But this composition is just a blast!
György Ligeti - Lontano You have probably heard some of his compositions in Kubrick films (2001: A Space Odyssey, The Shining - this one). So thanks to Kubrick stealing Ligeti's music (I mean using it without permission), we now have such audio-visual mastery that is 2001: A Space Odyssey. Well played.
Harry Partch - Exordium: The Beginning Of A Web (Delusion of the Fury) Well, there are no classical instruments here, but ones made by Partch himself. (although maybe he left some, I'm not sure)

What about any that incorporate interesting lyrics? Or artists within the realm of Xiu Xiu or the Velvet Underground (who aren't quite Avant-garde, but you get what I mean).

I'd look into the now recently departed Captain Beefheart for that. Like or no, There's no way you could not consider his lyrics interesting.

Yea... I could never bring myself to like him before. Sorry to hear that he's dead though.

I love Partch's instruments. I also love the vagrant theme of Delusion of Fury.

Honestly man, I don't know. You seem to be really trying to get into experimental but not liking it from the lines of questions.

Trying to help, but not much I can do. What do you like of the genre?

Xiu Xiu, Velvet Underground, Arthur Russel, John Cage, The Residents, Frank Zappa, and the Flaming Lips are a few avant-garde/experimental rock musicians. I know it's strange to like Zappa and not the Captain, though.

ah...my apologies. I thought you were struggling to get into the genre, and trying to figure out why everyone else loved it so much.

I'm blanking on a name... The first letter of his name starts with a "k." He's an avant garde composer who was very influential with Throbbing Gristle. Does anyone know?

Krzysztof Penderecki, perhaps? Not an 'avant-garde composer' by definition but some of his work kind of suggests it.