I want to quit my public school job. I don't want to be sued or anything. It says I need to give a 60 day quit notice, but I want to give a 30 day quit notice.

My plan is to give my 30 day notice. Since I have about 10 days of vacation leave, I will teach for 2 weeks and 2 weeks vacation.

My question is what is the legal actions the school can go after me if I quit? Also, can my public school go after me legally because of the students? They have to find a new teacher within those two weeks because no one can teach those classes?

Might be better to just do a run. This is your first year correct? So no severance and maybe no pension claim too. Just go.

I have a marriage visa. Does that make a difference?

Afterwards, get on a plane home, but make sure you go on first class so you can enjoy a celebratory glass of rye.

You'll be screwing over your school and coworkers if you give them 2 weeks notice or midnight run. Your coteachers and other staff will have to cover your classes and the school will be left dealing with all the loose ends you leave behind. In the longer term, it'll probably sour everyone's opinion on native teachers as well and cause problems for your eventual (if they opt for it) replacement. They won't approve your 10 days off mid semester so that you can leave at the 2 week mark instead of 30 days (really? lol). I don't think they will find a replacement native teacher in that short a time, it's possible they could get a temp Korean teacher though (my previous school struggled to find temps when needed as there was a shortage in the area).

My contract states that if you don't give 60 days notice they will report you to immigration. I wouldn't worry about legal action beyond repaying prorated vacation days (as Mr C suggested), and your flight allowance if you're quitting before the 6 month mark. I don't imagine this would be an issue unless you try to avoid repaying those. It will likely blacklist you from working for that office of education ever again, and EPIK as a whole if that was the path you took. If you live in a small town it may affect your shot at other local jobs as well.

If none of the above is of concern to you, then as the others said it is probably easier to just leave. Since you're married though I assume you will be either staying with your wife or (if leaving the country) returning eventually? If it's possible I would just do the 60 days. It would be better for your school and coworkers, though I think even then the education office and epik/gepik/whoever won't look kindly on you in the future.