Visa application takes a couple of days to spend abroad. It is not free, cost depends on consulates but it's around 1,
*****
,500 baht.
Consulates could also ask financial proof, proof of lodging (hotel booking or house rental contract), and a flight out of Thailand within 60 days.
Visa exempt is just a stamp that you can get with a same day go and back border run. The stamp is free. From Koh Tao you could go to Ranong border with Myanmar. You just need to pay a fee of 500 baht (or a brand new 10 US dollar banknote) to enter Myanmar.