Briefly, you give your school whatever documents they ask you for, and they process them.
You need to be tolerant and helpful at this point even if you personally think the school has no need to ask for certain things. If you don't co-operate, the process will get stuck and you'll be the one leaving the country to get a new visa.
First they obtain a teacher's license for you, and then they use this to get a work permit. Once you have a work permit, your visa can be extended.
Lengthily, (and the details may vary between schools and provinces) the first step is the teacher's license. This requires more or less the following:
A personal information document, up to 12 one and a half inch photos, up to 12 two inch photos, a current health certificate (50 baht any hospital - are you alive? yes - you passed), copies of your degree and other certs (originals may be requested along with transcripts) certified Thai translations of your degree and other certs, copies of every page of your passport, school director's license, school principal's license, map of school, teaching schedule of teacher, list of other work permit holders at the school, a new blank teacher's license book (blue), form Sor Chor 10, form Sor Chor 17, form Ror 11.
And if you've had a license before, then you can add Ror 12, Sor Chor 19, Sor Chor 18, and your blue license book (not to be confused with the work permit book, plus some provinces don't issue them, in which case you might need to fabricate a police report saying you've lost it)
These are all submitted in quadruplicate to the Thai Ministry of Education - and don't forget to sign every single page.
After a while (one week to who knows) you will get the license back. You take this, along with most of the same documents as above to the Labour Department, who will issue a receipt of application.
You can use the receipt to extend your visa - the implication being that your application won't be turned down at this stage.
After about three weeks you pick up your new work permit.
When the time comes for renewal, it's a good idea to remind your school about one month before the expiry date, though they really should be on top of things by that stage.