Many pet parents get their pet from a shelter or rescue center. Kudos to you if you've done that! Taking that next step of getting them insured can be tricky. Most insurers need to know the age and breed of your pet, and the adoption center may not know this information. The pet may also have a pre-existing medical condition that isn't apparent and most pre-existing conditions will eliminate your pet from being eligible for insurance coverage. There are things you can do to lift the cloud of a mystery though.
Take your pet to a trusted veterinarian for a thorough examination. They will be able to provide you with an estimate of their age and breed, and with a physical exam that may include some lab work, they'll be able to alert you to any pre-existing conditions or illnesses. You'll then be armed with the information you need when speaking with a licensed insurance agent about getting coverage for your pet.