I, too, cringe at "Me and my friends are going..." or "Me and her..." I sometimes reply, "Me is going to (whatever)." Sometimes it works. But I can't do it to the people on TV, where even the talking heads say it.
Further down the list, but related, is, "They (did whatever) to she and I." ...sometimes from the same people. The irony!
I'll be a middle school math interventionist when school starts this year. I've done reading interventions as well over the last five years at the elementary level. Before becoming a teacher I did graduate work in linguistics.
When they speak, I'm not disturbed by grammatical violations I hear my students commit. I hesitate to call them errors because so many are intentional. I really don't care what slang my students use. I know that their double negatives are supposed to communicate emphasis ("After what he did, Suzie ain't never going out with Bob again"). I don't mind when my kids regularize irregular verbs and tell me which ridgeline they were "borned" on. Dialectal variations of the second person plural (yuns, you-uns, ya'll, yous, yous guys, etc.) are simply colorful. And language is supposed to be colorful.
What does disturb me is when a student doesn't know how to move from the imformal sort of speech where all this takes place to the more formal approach to language that you seem to be calling proper grammar. I don't want my kids to speak properly all the time. I want them to move back and forth between formal and informal speech in a way that demonstrates that they understands the social demands of a situation. If they can't to this it disturbs me. And it disturbs me primarily because it means I've failed in an aspect of our most important task as educators: socialization.
...and the fact that things like "Me and my..." are being written into TV scripts is evidence that this "socialization" is being made much more difficult by our media. But at the rate it's going, "Me and my..." or "Me and her..." will eventually be socially accepted grammar at all social levels of this upcoming generation, so maybe we fogies need to "fuggetaboutit!" But me is afraid me can't.
My cringing grammar pet peeve is improper use of prepositions. "I had to get DOWN from the car, or I had to get OFF the car." If you're IN the car, then you get OUT of the car. I draw a picture for my students of a car and ask them where would I be if I was ON the car? Then I draw a stick figure on the hood or roof. Then I go on to say, "You're telling me you drive around town like this? Because that's how you get OFF the car." They laugh.
Another is SANGwich for sandwich, I know it's a funny word already, but it's not singing. If it's all you hear, then that's all you know...
Oh wait, one more. I TOOK my sister a bath. I don't understand? You can take a bath yourself, but you GIVE someone else a bath. I gave my baby sister a bath. (ahh, better)
AND double negative city. It's not just the kids either, yikes!!!
I cringe when I hear "irregardless" instead of "regardless".
I also try to get my students in the habit of asking, "MAY I use the bathroom", instead of "CAN" in addition to "me and him", "he and I", etc.
I find the biggest challenge is that proper grammar is not being used at home, therefore not reinforced. Unfortunately, kids are getting mixed messages with grammar. I actually had a student WRITE an essay using "text" lingo and did NOT see a problem. Times are changing....
Children can be watching! I offered an assignment to my students one year, offering them bonus points for any glaring grammatical or spelling errors they found in the newspaper or in signs around town. They seemed to enjoy the exercise and produced a variety of photos and clippings for class discussion.
I cringe when I hear "less" and "fewer" misused. There are "less" students in this class than that one. Less is when you cannot count "it" and "fewer" if you can. There are fewer students...... and there is less milk in this glass than that one.....
It also drives me crazy when people call their realtor......"re la tor". It is pronounced "real tor".