I was riding a Metro bus which stopped suddenly because a car turned into its path, do I make a claim against Metro, or against the other driver?

While overall Maryland and the District of Columbia have very safe transportation systems the sheer number of people using them and the nature of the buses and trains utilized results in catastrophic injuries on a fairly routine basis.
These claims require detailed investigation in order to determine how best to pursue your claim.  Even if it appears that the conduct of the bus driver was appropriate from your observation, it may well be that their violation of Metro rules and regulations or ordinary rules of the road would permit you to make a successful claim against Metro.  Such cases require a timely and thoughtful analysis in order to be pursued properly.