During the 80s Australian bands largely ignored the overseas trend towards glam rock and produced lots of quality garage music. They played plenty of quality music that is largely little known overseas. Shock records have this compilation which starts with seminal bands Radio Birdman and The Saints.
If you are looking to educate your listeners and introduce them to something they probably haven’t experienced, well check this out.
I love all of the bands that have been mentioned so far.
Most of my would-be suggestions have already been mentioned, except for Killing Joke (their 90s stuff is more “industrial” in style, but their 80s stuff is very punk-inspired).
I don’t think that you can properly cover this era and genre of music without adding the 3rd wave ska folks - The Specials, English Beat, The Selector, Madness, Bad Manners, Hepcat, Bodysnatchers, Fishbone, the Untouchables, the Pietasters, etc. The music may not be “punk” per se, but the same crowd was generally found at both shows.
I haven’t seen The Cult mentioned, and Love is one of the best albums of that decade IMHO. Some others off the top of my head (mostly 80s alternative): Hoodoo Gurus, Camper van Beethoven, REM, The Connells, Big Country (an incredible band unfortunatly only really known for one song), Sinead O’Connor, Adam Ant, The Alarm, Bad Religion, Echo & the Bunnymen, Fear, fiREHOSE, Generation X, INXS, Let’s Active, Living Color, Midnight Oil, Split Enz, Squeeze, Jesus Mary Chain, and for some real fun, the Toy Dolls.
For good (hell, make that excellent) reference material, try to dig up any of the Trouser Press guides. IIRC, there are four editions, but any of them can work for you. They’re out of print, so you best bet is to start looking in used book stores.
Another punk act from the late '70’s:
The Contortions: Buy the Contortions ('79). Perhaps best described as “art-noise punk-jazz”. If you like Dick Hell & the Voidoids, Nick Cave & The Birthday Party, and The Lounge Lizards, you’ll love these guys. For one thing, they had a guitarist who was a Robert Quine sound-alike, and a rapid-riffing saxophonist…
A few relatively obscure items to look for from the early '80’s:
The Birthday Party [prev. the Aussie band Boys Next Door, both with Nick Cave]: **Prayers on Fire ** and **Drunk on the Pope’s Blood ** (both '81).
The Vapors: **New Clear Days ** ('79) and Magnets ('81), since consolidated on one budget CD under the title Vaporized. Widely dismissed as one-hit wonders (for “Turning Japanese” from NCD), their albums are suprisingly good.
XTC: Explode Together (The Dub Experiments, '78-'80) (released on CD under that title in '90). Andy Partridge took tracks from their first three LPs and smashed them together, with lots of distortion, compression, added dub rhythm tracks, etc. Musically, it’s really for XTC fans only, who can have fun identifying the musical bits, but this album anticipated trends in electronica/techno, 80’s ska revival, and the eclectic turntablism of artists like DJ Danger Mouse (The Grey Album).
The Golden Palominos: **The Golden Palominos ** ('83). This LP is truly unique (and sounds nothing like the rest of the GP’s discography); perhaps best classified as “SoHo art-noise jamming”. Particularly recommended for fans of percussion and the NYC arts scene.
Corrections: Elvis Costello’s **Armed Forces ** came out in '79, and I overlooked his **This Year’s Model ** from '78.
Part of the reason you wouldn’t is that you can’t:
I’d say that some of the genre-defining punk bands are:
The Exploited
Dead Kennedys
Minor Threat
Black Flag
Fear
Circle Jerks
Suicidal Tendencies
GBH
JFA
The Misfits
The bands served different roles. For example, Minor Threat defined D.C., straight-edge hard core; The Exploited, so I was told, were a band that straddled punk & oi!, and played a hard, fast, British punk—foot-tall mohawks, bondage pants and Dr. Martin’s would indicate that a person probably had an Exploited album or two; Circle Jerks, Suicidal Tendencies, & JFA (Jodi Foster’s Army), IIRC, were what would become skateboarding punk music; The Misfits put satan in punk and their fans looked the part.
Watching The Decline of Western Civilization would be good for you, I’m sure. 
Ahhh my personal favorite section of music.
Here are my reccomendations:
New Wave:
The cure - Pornography, Disintigration, Head on the door
Depeche Mode - Music for the masses and Black Celebration (there are also 2 cd’s of singles our from just their 80’s years)
XTC - Skylarking, Oranges and Lemons, English Settlement
Bauhaus - Swing the Heartache (or there are 2 great compilation CDs)
Oingo Boingo - Anthology has all their great stuff on it but if you don’t want best of’s i reccomend “nothing to fear”
Echo and the Bunnymen - pretty much any album
Romeo Void - can’t recall the album name but it has the song Never say Never
Fishbone - Truth and Soul or the self titled EP.
The Smiths - Meat is Murder, Strangeways here we come, Louder than Bombs, The Queen is dead
The The - Infected and Mind Bomb
Not really new wave - The pogues - hell’s ditch
Underworld (not the elctronic stuff the 80’s stuff) - Underneith the radar, change the weather
Fruer - doot doot
Peter Schilling - pretty much just the song “major tom”
David Bowie - Let’s Dance
Men at work - business as usual
Adam ant - dirk wears white socks, friend or foe, kings of the wild frontier
punk:
The Dickies - Stukas over disneyland
Descendants - Milo goes to college
The Misfits - Collection 1 and Collection 2 has all their best stuff
The Vandals - Fear of a Punk Planet
Dead Kennedys - Give Me Convenience or give me death
7 seconds - Walk Together Rock Together
The Adverts - pretty much any album
GBH - any album
MDC - millions of dead cops
Black Flag - Wasted, damaged
Iggy Pop - Lust for Life
Circle Jerks - group sex/Wild in the Streets
Social Distortion - self titled, somewhere between heaven and hell, mommy’s little monster, mainliner
THe Ramones - rocket to russia, road to ruin
The Stooges - self titled
Operation Ivy - self titled
The Cramps - songs the lord taught us
Angry Samoans - back from samoa
A good collection of stuff is on some soundtracks… like pick up the soundtrack to SLC Punk or Repo man, they have good collections
If you want more i can give you more
Sitting somewhere between New Wave and Punk don’t forget Industrial (especially Wax Trax! stuff). I didn’t see it mentioned previously, so here goes:
Big Black (Rich Man’s 8-track Tape is best IMO, the big hit on this album was the song “Kerosene” although the my college station always played “Jordan” whenever I called in) for those who appreciate the punk.
Ministry (With Sympathy can go on your regular New Wave show and Twitch would work there as well although it’s a bit darker. Land of Rape and Honey (especially “Stigmata”) and The Mind is a Terrible Thing to Taste (especially “Burning Inside” and “Thieves”) would go well with the punk part of the show).
For some reason KMFDM always appeals to nonrivetheads. I have no idea why.
I highly recommend looking into those bands and albums for your show.
I agree with the industrial and goth … I vote for:
Skinny Puppy “Smothered Hope”
Ministry “Stigmata”
Sisters of Mercy “Lucretia (My Reflection)”
Depeche Mode “Fly on the Windscreen”
Erasure “A Little Respect”
What do you mean one of these doesn’t fit in?
The Cult occupied an important space in the early 80’s music scene and bridged and evolved through both the punk and new wave genres.
Originally known as The Southern Death Cult they were a decidely punk influenced band, having initial success in 1982 England with a self titled album and the singles Moya and Fatman. In 1983 they truncated the name and became The Death Cult (God’s Zoo, Brothers Grimm, Ghost dance, Horse nation). They released the album Dreamtime in 1984 with their final, yet again minimalized, name, The Cult, and had a #1 hit in the single Spiritwalker. This period of music solidly lodged them in the punk category. They changed direction with their next album and first international success, 1985’s Love (She Sells Sanctuary, Rain, Nirvana), this album is usually categorized as New Wave and definitely was a 180 degree change in direction from their Punk roots. Billy Duffy, the Cult’s guitarist also came from the early 80’s English punk scene originally playing for the punk groups The Nosebleeds and Theatre of Hate.
Heartily seconded. Radio Birdman are awesome. Sometimes called “pre-punk,” a lot of their stuff anticipates punk music, but some also harks back to the rhythms of surfer music, and there are places where they show a definite Doors influence (the song “Man with Golden Helmet” comes to mind).
I’d also like to throw in a vote for the LA band X. They toured last year, and i saw them when they visited Baltimore. Despite the fact that they’re (almost) old enough to be my parents, they still rock.
Someone else mentioned the British band The Buzzcocks, who are also great. Saw them here in Baltimore last year, too. If you want to play one of their songs that non-punk fans will recognize, try “Ever Fallen In Love (With Someone You Shouldn’t 've),” which was covered by the Fine Young Cannibals on their 1989 album The Raw and the Cooked.
No one’s mentioned the Rezillos or the Revillos? How about Hayzee Fantayzee? X-Ray Spex?
Y’all ain’t tryin’.
That’s one of my favorites too, although i think that the previous album, Red Roses For Me, also has some rollicking good stuff. “Waxie’s Dargle,” for example, shows great punk energy.
“Shiny Shiny”? Give me a break. That song, like Klaus Nommi’s “Total Eclipse” only work in video format, where the kitsch can be seen and duly appreciated. Just hearing the songs is boring (much like watching Rocky Horror Picture Show at home on DVD.
I’d just like to pipe up with Mission of Burma. The songs “Acadamy Fight Song, That’s When I Reach For My Revolver, and That’s How I Escaped My Certain Fate” are personal favorites.
For the Pogues, don’t forget “If I Should Fall From Grace With God” The energy was still there for this one (especially Bottle of Smoke and Medley), but the production values are a bit better. Don’t bother with Peace and Love or anything after.
So far, I’ve been able to find the Sex Pistols’ “Never Mind the Bollocks” and “London Calling” by the Clash. I’ve also got some random songs; not much, but enough to get me through the first week or so.
I need to scope out the local used-record store for the rest of these.
Robin
I’ll suggest the B52’s and Wild Planet.
B-52s first album 1979
B-52’s Wild Planet 1980
I was composing this post when the hamsters exceeded critical mass and imploded. 
C’mon down to my part of town-there’s a huge used CD/music store in York. See if there’s anything you need to raid from my library, too.
You can never go wrong with the Dickies.
Check out Dawn of the Dickies. Stukas over Disneyland has already been suggested.
But, but, but… it’s in York!!
Seriously, some day when I’ve got some time (heh) AND money on the same day, I’ll come out, we can do lunch, then do the music store.
Robin