Further recommendations for your exploration: any band Ian MacKaye has played in. Even if you don't care for his work, he has had an outsized impact on the development of American punk / hardcore (even if he's resisted that attribution). Specific recommendations include:
- Minor Threat - self titled (straight up hc punk)
- Embrace - Building (early emo, when that just meant it was a hardcore song that touched on emotions other than anger)
- Fugazi - Waiting Room, or the 13 Songs album as a whole (post-hardcore, or "hardcore, but we're not afraid to show off that we've learned some music theory")
Okay, with MacKaye given his due, other "classic punk" recommendations include:
- Agent Orange - Bloodstains
- Bad Brains - I Against I
- Black Flag - Fix Me, TV Party, My War
- Circle Jerks - Wild in the Streets, Live Fast Die Young, World up My Ass
- Dead Kennedys - Nazi Punks Fuck Off, California Über Alles, Holiday in Cambodia
- Germs - Lexicon Devil
- The Clash - Straight to Hell, White Riot, Lost in the Supermarket, and many more.
- Operation Ivy - Sound System
- The Descendents - Suburban Home
- Social Distortion - Story of My Life, Mommy's Little Monster
- Buzzcocks - Ever Fallen in Love (With Someone You Shouldn't've)?
- Hüsker Dü - Don't Wanna Know If You Are Lonely, Pink Turns to Blue
- The Replacements - Androgynous, Bastards of Young, Here Comes a Regular, Can't Hardly Wait, Answering Machine, Alex Chilton
- Minutemen - History Lesson Part 2, Corona
Some slightly deeper cuts, more in the proto-emo space than political punk, but they share a lot of musical DNA and I think some of these tracks are underrated.
- Rites of Spring - For Want Of
- The Hated - Words Come Back
- Dag Nasty - Circles
- Gray Matter - Burn No Bridges
- Soulside - Pearl to Stone
- IGNITION - Previous
- Fire Party - Cake
- Samiam - Tired of Waiting
- Fuel - Cue to You
- Drive Like Jehu - Here Come the Rome Plows
That's probably enough for now. I hope you find some stuff you like here.