There are all kinds of different health insurance, and you have to pick what works best for you. Most people who have insurance available through their employer take advantage of that option as it is usually, but not always, better insurance for less money. Also, employer provided health care plans sometimes provide tax benefits also. You need to get information on plans that are available and compare them.
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You are very smart to be thinking about this important topic. You can obtain your own private insurance. Try to find a well known company such as Blue Cross Blue Shield, United Health Care, Aetna, Cigna, and avoid major medical policies that provide minimal coverage. I have Blue Cross Blue Shield and BCBS has retail locations which are convenient. Being insured is very important in this country. Good luck!