Inclusion Criteria:

          1. Is a male or female participant of 18 years or older.

          2. Has histologically confirmed, unresectable Stage III or Stage IV melanoma per the
             American Joint Committee on Cancer (AJCC) staging system.

          3. Has an eastern cooperative oncology group (ECOG) performance status of 0-1.

          4. Adequate bone marrow reserve and renal and hepatic function within 28 days before the
             first dose of study drug on the basis of the defined laboratory parameters.

          5. For TAK-580 + nivolumab and TAK-202 (plozalizumab) + nivolumab only: Had disease
             accessible for repeat nonsignificant risk biopsy (those occurring outside the brain,
             lung/mediastinum, and pancreas, or obtained with endoscopic procedures extending
             beyond the esophagus, stomach, or bowel) and willingness to undergo serial tumor
             biopsies.

          6. Additional Inclusion Requirements for TAK-580 + nivolumab

             a) BRAF V600 mutation-positive or NRAS mutation-positive disease previously untreated
             with RAF, MEK, or other inhibitors of the mitogen-activated protein kinase (MAPK)
             pathway. Participants who have progressed on these agents can still be enrolled in
             TAK-202 (plozalizumab) + nivolumab or vedolizumab + nivolumab + ipilimumab.

          7. Additional Inclusion Requirements for expansion cohorts only a) Measurable disease per
             Response Evaluation Criteria in Solid Tumors (RECIST) guidelines (Version 1.1) and at
             least 1 nonsignificant risk, non-target lesion accessible for biopsy per the
             guidelines above (for TAK-580 + nivolumab and TAK-202 (plozalizumab) + nivolumab
             only).

        Exclusion Criteria:

          1. Has active brain metastases or leptomeningeal metastases. Participants with brain
             metastases are eligible if these have been treated and there is no magnetic resonance
             imaging (MRI) evidence of progression for at least 4 weeks after treatment is complete
             and within 28 days prior to first dose of study drug administration. There must also
             be no requirement for high doses of systemic corticosteroids that could result in
             immunosuppression (greater than [>] 10 milligram per day [mg/day] prednisone
             equivalents) for at least 2 weeks prior to study drug administration.

          2. Completed a prior therapy less than (<) 2 weeks prior to first dose and for whom
             adverse events (AEs) related to prior therapy had not returned to baseline or improved
             to Grade 1.

          3. Has active, known or suspected autoimmune disease.

          4. Has a condition requiring systemic treatment with either corticosteroids or other
             immunosuppressive medications within 14 days of study drug administration.

          5. Has a history of pneumonitis requiring treatment with steroids; history of idiopathic
             pulmonary fibrosis (including pneumonitis), interstitial lung disease, drug-induced
             pneumonitis, organizing pneumonia, or evidence of active pneumonitis on screening
             chest computed tomography (CT) scan; history of radiation pneumonitis in the radiation
             field (fibrosis) is permitted.

          6. Is previously diagnosed human immunodeficiency virus (HIV) infection or active
             hepatitis B or C.

          7. Additional Exclusion Requirements for arm 1 only (nivolumab Plus TAK-580)

               1. Concomitant use or administration of clinically significant enzyme inducers less
                  than or equal to (<=) 14 days before the first dose of TAK-580.

               2. Treatment with gemfibrozil (or other strong CYP2C8 inhibitor) within 14 days
                  before the first dose of TAK-580.

               3. Left ventricular ejection fraction (LVEF) <50 percent (%) as measured by
                  echocardiogram (ECHO) or multiple gated acquisition scan (MUGA) within 4 weeks
                  before receiving the first dose of study drug.

               4. Known gastrointestinal (GI) disease or prior GI procedure that could interfere
                  with the oral absorption or tolerance of the TAK-580.

          8. Additional Exclusion Requirements for arm 3 only (vedolizumab Plus nivolumab Plus
             ipilimumab)

               1. Had prior exposure to rituximab, natalizumab, vedolizumab, or alemtuzumab.

               2. Has a history of any major neurological disorders, including stroke, multiple
                  sclerosis, or neurodegenerative disease.

               3. Has taken any live vaccinations within 30 days before study drug administration
                  except for the influenza vaccine.