In deciding who to sue, bear in mind that, most often, doctors are independent practitioners and not hospital employees. That is why the attending doctor is usually the one who should get sued for medical errors. Sometimes you can also sue other attending specialists.
In most medical negligence cases, the doctor who treated the patient ought to be the one sued for misdiagnosis or wrong treatment. There are also circumstances where other medical specialists who were involved in the case may be sued for the suffering added to the patient. These also include the lab individuals who performed tests on the patient and various specialists participating in the case.
It is unusual to have the hospital, as an institution, brought into malpractice negligence cases, because most doctors are not hired directly by the hospital. Therefore, the hospital can't be responsible for a misdiagnosis error or any medical mistake made by a specialist.