Just walk into the drugstore and ask for the medicine you need and they will give it to you without a recipe. The sign can be recognized easily. You may even be able to memorize it from the picture. If not, I believe you will recognize it again if you see it in Thailand. There are lots of drugstores and you will see this sign very often.
Medical care is excellent in Thailand.
Like at home, doctors in Thailand are well educated and trained. They will always speak English and normally they will have a small private clinic, where they also can do simple laboratory experiments. They will normally also sell medicines. Such a clinic may look like the one on the picture above. If in doubt, simply enter and ask if there is a doctor working there.