As with any other situation where someone else's negligence played a role in causing an accident (and your resulting injuries), if you were riding a bicycle and were hit by a car, you can file a personal injury lawsuit in the local branch of your state's civil court.
But that doesn't mean you should rush to the courthouse steps. As long as you pay attention to time limits for filing a lawsuit (what's known as the statute of limitations), going to court will remain an option for you if you've been injured in a car-bicycle accident. But it's a costly and time-consuming option, so it's one you want to keep in your back pocket for now.
You can get compensation for your injuries by filing a claim with the driver's car insurance carrier, and reaching an agreeable settlement, without ever having to go to court.
This is called a "third party claim," and in most states you'll be able to recover compensation for your medical treatment, any lost income, repair/replacement of your bicycle, and your pain and suffering in connection with the accident and your injuries. It's the manner in which most personal injury cases are resolved.
Your best first steps will probably be to:
(See this sample demand letter to see what it might look like and what to include.)
Even in a car versus bicycle accident, where you've got a 5,000 pound vehicle going up against an exposed bicycle rider, the rules of the road still apply. There are situations where a bicycle rider comes away from an accident with severe injuries, but he or she also bears most of the fault for causing the accident.
If you were on your bicycle and were hit by the driver of a car, that person's car insurance will probably be the best first place to turn for making an injury claim. Bicycle riders face unique challenges when making injury claims with car insurance carriers.
This is all not to mention the fact that if you were the victim of a hit and run where the car driver didn't stop, your legal options will be seriously limited. You'll likely need to first turn to your own health insurance coverage to pay your medical bills, and then hope that you (and local law enforcement) are successful in your effort to locate the at-fault driver.