Yes, most state laws require that when you move and take up residence in a new state that you transfer over your registration, title and insurance for your vehicle to the new state and also exchange your driver's license from your previous state for one from the state you moved to.
Many states have a time period in which you must change over your license, title and registration, the normal amount of time being 30 to 60 days. If you do not change over your car registration, license, etc during the time period given then you may be cited for driving without registration or without license.
You likely cannot keep your insurance from your previous state whether you change over your driver's license to your new state or not. Typically when you register a car in a state you will also have to provide proof of insurance from a company authorized to do business in that state.
Also most insurers would not want to keep insuring you if you and your vehicle are now residing out of state. The garaging location of the vehicle is important to an insurance company and once you move into a new state they will normally advise you to get insurance now from that state, especially since state financial responsibility laws / insurance laws differ from state to state.
For example the NYS DMV states that you must apply for a NYS driver license not more than 30 days after you become a NYS resident. You also have 30 days after you become a resident to register your vehicle in NYS. When you apply for a New York title and registration you must have valid NYS liability insurance. You cannot use proof of insurance from another state under any circumstances according to the NYS DMV.
In Colorado you must apply for a Colorado title and registration for your vehicle within 30 days from establishing Colorado residency. You also must obtain a CO driver's license in a short period of time. You must provide proof of auto insurance to your county clerk and recorder before registration can occur.
You can contact the Department of Motor Vehicles in your new state for information on how long you have to change over your driver's license, registration, title, etc. The DMV or the state's insurance regulatory body can also tell you about their auto insurance requirements and if they permit your insurance to be from an out of state insurance company or if you must you an insurer from in state.