If you think that a local government entity (or a government employee) is to blame for an accident, you don't need to hire a special kind of lawyer in order to bring a legal claim. You're most likely looking for a personal injury lawyer who represents people who have been injured as a result of someone else's negligence. But you might want to consider hiring a personal injury attorney who has experience bringing similar claims against the government, maybe even against the same municipality that you're looking to hold responsible for your injury.
From a procedural standpoint, the kind of case you're talking about is not a "lawsuit"—at least, it won't start out as one. You cannot usually just file a lawsuit against a government agency or government employee after an accident, the way you would a private citizen or business.
Cities, counties, and other governments are free to set the rules under which they can be held liable for causing injuries or property damage. As a result of that freedom, most municipalities have put strict procedures in place for a claimant to follow before he or she can get compensation for an injury. You usually need to file a claim within a short amount of time after your accident or injury—30 to 180 days in most cases.
What do you need to include in the claim (also called a "notice of claim" in some places)? The answer depends on the rules in place in the city or county. The government might have its own claim form for you to complete, or they may just maintain a list of required information. The claim procedure may not seem especially complicated, but it's crucial to comply with all applicable deadlines and filing requirements. Once you've properly submitted a claim, the government will have a certain number of days (set by statute) to either deny your claim, settle it, or take no action at all. If your claim is denied or no action is taken, you're usually free to file a lawsuit. So in this way the claim process acts as a sort of prerequisite to taking the matter to civil court.
Your best first step, if you think a government agency might be responsible for your accident injuries, is to contact the clerk's office for that agency and ask about the process and requirements for filing an injury claim. You can also start by doing an online search using a phrase like "claim against [name of state/city/county] government."
Or, if you're fairly certain about the government's liability, and your injuries are pretty serious—you were in a car accident where a city bus ran a red light and hit you, for example—you may want to contact a local attorney right away. Chances are, you'll be able to discuss your case and your options without paying any up-front fees. Learn more about finding the right personal injury attorney for you and your case.