I am uninsured and was hit by an insured driver. Should I contact their insurance or would I get in trouble even if they're at-fault? 
There are two issues here . First , you state that you do not have car insurance yet were driving . I can only assume that the car you were driving is your own , registered to you . In a state like Connecticut , where it is required by law to carry car insurance in order to register and drive a car on public roads and highways , if a police officer was on the scene , you would have been issued a ticket . and possibly would have had your car towed . The second issue is that you wer hit by another car . That person , in the absense of any other details , would probably be held at fault for the accident and therefore their insurance company would pay for the damage repair to your car . 