In addition to buying a home, applying for Social Security number, opening bank accounts, getting a driver's license, buying a car insurance policy, and maybe getting a credit card, a newly arrived immigrant will need health insurance almost immediately. If you are in ill health, you need to plan and time this very carefully because you may get into a lot of trouble if you need medical assistance.
Most American residents get health insurance coverage through their employer so if you have a decent, full time job, you will be covered by a policy from your employer (discuss this with your employer at the time of interview and then you will asked to do the paperwork on the first day of work). If you are going to be without a job for whatever reason, you can buy insurance through the health insurance exchanges run under ObamaCare. They can be more expensive but are the only option for those who work part time or are self employed or simply cannot work.
In the meantime, if you get seriously sick, you can visit a hospital emergency room but expect to get a big bill in the mail later on. Do not go there if the illness is not urgent or life threatening because without health insurance, you maybe denied care and sent home with a suggestion to see a regular doctor. There are some doctors who see patients without insurance (and charge in cash) and you can find information about them online or through people you might know, particularly in the immigrant community. These doctors are typically general physicians and can only provide treatment for minor diseases but for peace of mind and for major health issues, you will definitely need health insurance.