This is the best and biggest hospital in Cheonan and for anything serious, this should be the place you go. The American military gets people treated here so they're used to working with foreigners and most people speak at least a bit of English. They have an international clinic for general stuff, as well as more specialized clinics for feet, pain, obesity, etc. They do the full range of surgeries as well. But prices are not cheap like the little neighborhood clinics, it's about 15 000 Won for a basic consultation. Here's the website.
Here's the link
with important phone numbers.
I recently got a complete checkup done there for about 400 000 Won and was extremely satisfied with the service. Very well organized and comprehensive. It was outrageously cheap for the amount of tests that were done. Here's the site for that