This is the range of dates within which you can enter Thailand. You could then stay 60 days, and extend that by another 30 days at an Immigration Office. At that point you would need to leave Thailand for a period (an hour/day) and get a 60 day visa-free stamp when you come back in. You can then extend that by another 30 days, as before. Rinse and repeat, until a border official thinks you may not be a genuine tourist
Busiest days in Immigration Offices are Monday and Friday, so avoid them. And if you get there early (check opening times for the one nearest where you'll be staying) you can easily be out in a couple of hours. Take 1,900 cash, your passport and a photo