The CDC, under the Health and Human Services Secretary, has some authorization to do quarantines (emphasis mine)
Under section 361 of the Public Health Service Act (42 U.S. Code § 264), the U.S. Secretary of Health and Human Services is authorized to take measures to prevent the entry and spread of communicable diseases from foreign countries into the United States and between states.
The CDC can do a state-level quarantine. Let's say Rhode Island becomes 100% infected. The CDC could prevent citizens of Rhode Island from traveling anywhere outside Rhode Island. They could not enforce it at the local level, since they lack jurisdiction.
States have police power functions to protect the health, safety, and welfare of persons within their borders. To control the spread of disease within their borders, states have laws to enforce the use of isolation and quarantine.
These laws can vary from state to state and can be specific or broad. In some states, local health authorities implement state law. In most states, breaking a quarantine order is a criminal misdemeanor.
State and Local
At this point, the HHS secretary and/or the CDC would ask Rhode Island's governor for help from within the state. Since we picked Rhode Island, we need to look at the state laws
§ 23-8-4 Quarantine. – If the state director of health, or his or her duly authorized agent, determines, upon investigation, that a threat to the public health exists because any person is suffering, or appears to be suffering, from a communicable disease, the director or his or her authorized agent may require or provide that person to be confined, in some proper place, for the purpose of isolation or quarantine, or another less restrictive intervention treatment, including, but not limited to, immunization, treatment, exclusion or other protective actions until the threat to the public health has abated. Nothing in this section shall be construed to prevent a person who is unable or unwilling for reasons of health, religion, or conscience to undergo immunization or treatment from choosing to submit to quarantine or isolation as an alternative to immunization or treatment. Orders under this chapter shall be in accordance with the procedures for compliance order and immediate compliance orders set forth in §§ 23-1-20 – 23-1-24. A person subject to quarantine under this section shall be entitled to file a petition for relief from such order at any time, included, but not limited to, a petition based upon compliance with a treatment under less restrictive alternatives.
Most states likely have similar laws to prevent the spread of disease. At this point, the governor would either contact the proper state authority, or invoke whatever laws were needed (governors tends to have broad emergency powers and could probably unilaterally declare a state of emergency) to ensure the people of Rhode Island weren't just walking around on the streets. Since all state law enforcement tends to report to the governor anyways, this would become a directive to all state and local law enforcement.
Limit public meetings
This one is a bit trickier, but it's still possible with proper emergency declarations to ensure that large public gatherings are prohibited. In Maryland, Governor Brian Hogan issued a State of Emergency on March 5, and today announced several new restrictions since they have a confirmed case
Maryland Gov. Larry Hogan said Thursday that he is banning all public gatherings of 250 people, activating the National Guard, ordering all nonessential state government employees to work from home and closing the Port of Baltimore to passenger cruise ships.
“This is a public health emergency,” Deputy Health Secretary Fran Phillips said.
It's important to remember that Hogan is a Republican governor in a largely Democratic state. As such, if the legislature disagreed with his declarations, they could pass laws to revoke the state of emergency, or remove the governor and have his successor remove the restrictions and/or state of emergency.
Why isn't this a violation of the First Amendment? Because it's viewpoint neutral
Viewpoint discrimination occurs when a governmental regulation restricts expression based not only on its content, but specifically on the underlying views in the message.
Content-based restrictions limit speech based on its subject matter, while viewpoint-based restrictions limit speech based on ideology and perspective. A law banning all political speeches in a public park would be content based; a law banning only political speeches by members of the Socialist Party would be viewpoint based. In the words of Justice Anthony M. Kennedy in Rosenberger v. Rector and Visitors of the University of Virginia (1995), “Viewpoint discrimination is thus an egregious form of content discrimination.”
In other words, it doesn't matter why you have more than 250 people, only that you have that many, and it is prohibited by emergency declaration.