Study to Evaluate the Safety, Tolerability, and Pharmacodynamics of Investigational Treatments in Combination With Standard of Care Immune Checkpoint Inhibitors in Participants With Advanced Melanoma

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.