Most lawyers take personal injury cases on a contingency fee basis. In most instances the usual practice is for the lawyer to take a fee of one-third out of the settlement for a verdict received in the case. Many, but not all lawyers, will pay for the out of pocket expenses associated with pursuing the claim. Some lawyers may ask the client to front the lawyer the money for expenses. In that instance if the case is lost, then the client loses those funds.
Most lawyers can tell a client in advance if the client has to pay expenses up front.
Otherwise, the case should not cost the client any money out of pocket. The answer is yes. Claims are brought against the person whose carelessness caused your injury. If that person was in the course of their employment at the time, then you would probably bring a claim against their employer as well. In today's society most individuals and businesses have insurance.