After you have signed the papers from your school, they will send them to the ministry of education for approval. When approved, they will send them back to your school. With the documents in hand, your passport and some passport pictures you are ready to apply for the visa.
Very few language schools can get your tourist visa/visa exempt converted to an ED visa inside the country. Duke in BKK is one of the exceptions, but they don't have a school in Phuket, so they are of no use to you.
I do not know if any schools here in Phuket can convert inside the country, but you can ask about it in a Phuket forum.
If you can convert inside the country, you just go to the local immigration and hand in your application.
If you need to convert outside, you will have to travel to an embassy outside the country to apply for the visa (you will have to stay 1 or 2 nights)
I can see the upsides of using an agent for certain types of visas or tasks, but there are no benefits of contacting an agent when obtaining a genuine ED. Find a school and take it from there. 🙂