Gemcitabine Hydrochloride, Cisplatin, and Nab-Paclitaxel in Treating Patients With Advanced or Metastatic Biliary Cancers

Inclusion Criteria: * Patient must have histologically or cytologically confirmed intrahepatic cholangiocarcinoma, extrahepatic cholangiocarcinoma, or gallbladder cancer or may undergo a repeat biopsy for histologic confirmation if pre-existing biopsy is not sufficient for diagnosis * Metastatic or unresectable disease documented on diagnostic imaging studies * May not have received prior chemotherapy; if patient has received prior adjuvant therapy, must be \> 6 months from treatment * Eastern Cooperative Oncology Group (ECOG) performance status of 0 to 1 * Absolute neutrophil count (ANC) \>= 1,500 cells/mm\^3 * Platelets \>= 100,000/ul * Hemoglobin \> 9.0 g/dL * Total bilirubin =\< 1.5 mg/dL (in patients with known Gilbert's syndrome direct bilirubin =\< 1.5 x upper limit of normal \[ULN\] will be used as organ function criteria, instead of total bilirubin) * Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) = \< 5 x ULN * Creatinine =\< 1.5 gm/dL * Negative serum or urine pregnancy test in women with childbearing potential (WOCBP) defined as not post-menopausal for 12 months or no previous surgical sterilization, within one week prior to initiation of treatment; WOCBP must be using an adequate method of contraception to avoid pregnancy throughout the study and for up to 12 weeks after the last dose of study drug to minimize the risk of pregnancy * A male subject of fathering potential must use an adequate method of contraception to avoid conception throug…