A good place to start is HealthCare.gov. This is the health insurance exchange created by the The Patient Protection and Affordable Care Act, and is a one-stop shop for private individual market health insurance plans (note that the exchange itself is run by the government, but the health plans for sale in the exchange are all private, from the health insurance companies with which you're already familiar). People in 39 states use HealthCare.gov to enroll in individual market plans. The other 11 states and the District of Columbia have state-run exchanges, and you'll be directed to their sites from HealthCare.gov when you select your state.
Individual and family health insurance plans can help cover expenses in the case of serious medical emergencies, and help you and your family stay on top of preventative health-care services. Having health insurance coverage can save you money on doctor's visits, prescriptions drugs, preventative care and other health-care services. Typical health insurance plans for individuals include costs such as a monthly premium, annual deductible, copayments, and coinsurance.