What steps should we take to cancel our visa and work permit after resigning from our teaching position in Thailand?

Are you on the original visa you purchased outside of the country, or are you on an extension of stay you applied and paid for from immigration? That makes all the difference.

If you are on the original 90 day visa, visas cannot be cancelled and you can stay in until your arrival stamp based on that visa runs out.

If you are on an extension of stay, your permission to stay in Thailand ends on the day your employment ends, because your permission to stay is based on that employment.

You must cancel your work permit with documentation from your employer and then cancel your extension of stay with the same documentation and your cancelled work permit.

You can do this 1-2 weeks ahead of time and they will stamp you in until the date of cancellation that is listed on your documentation.

If you already resigned and are not worked and were on an extension, then you're already in the country illegally on overstay and you should leave as soon as possible.

If you do not cancel your extension based on work, you will not be able to get a new one for a new job, and you'll be fined 20,000 baht for failing to do so because you signed a document stating you would information immigration if your status changed which it did when you quit your job.

You'll probably need to leave Thailand in all cases and apply for a new non-B visa outside the country to start the process over again.