Primary Objective To determine whether initial treatment with either combination ipilimumab + nivolumab (with subsequent dabrafenib in combination with trametinib) or dabrafenib in combination with trametinib (with subsequent ipilimumab + nivolumab) significantly improves 2 year overall survival (OS) in patients with unresectable stage III or stage IV BRAFV600 mutant melanoma Secondary Clinical Objectives * To evaluate the impact of initial treatment on median OS and Hazard Ratio for death. * To determine whether initial treatment choice significantly improves 3 year OS. * To evaluate the anti-tumor activities (RECIST-defined response rate, median PFS) and safety profiles of ipilimumab + nivolumab and dabrafenib-trametinib in a Cooperative Group trial of patients with V600 mutant melanoma * To evaluate the activity (RECIST-defined response rate, median PFS) and safety of dabrafenib + trametinib in patients who have had disease progression on ipilimumab + nivolumab and in comparison to its activity and safety in ipilimumab + nivolumab nave patients. * To evaluate the activity of ipilimumab + nivolumab (RECIST-defined response rate, median PFS) and safety in patients who have had disease progression on dabrafenib + trametinib and in comparison to its activity and safety in dabrafenib + trametinib nave patients. * To assess the feasibility of crossover to the alternative treatment strategy (percentage of patients who are able to cross-over from one arm to the other and complete at least an initial course (12 weeks) of treatment after crossover without intervening Secondary Laboratory Objectives Association of inherited variation with immune mediated adverse events and response to ipilimumab + nivolumab To determine the utility of circulating BRAF levels in determining the response and resistance to either BRAF/MEK directed and/or combination immunotherapy in patients with BRAF mutant melanoma Secondary Patient Reported Outcomes Objectives To evaluate differences in overall health between initial treatment arms (dabrafenib + trametinib vs. ipilimumab + nivolumab immunotherapy) at 2 years, accounting for toxicities and overall survival. (Primary) To assess differences in overall function over 2 years between initial treatment with dabrafenib + trametinib vs. ipilimumab + nivolumab (Secondary) To document the effects of treatment crossover and treatment administration sequenceon symptom burden and overall function(Secondary) Exploratory Tobacco Use Objectives To determine the effects of tobacco, operationalized as combustible tobacco (1a), other forms of tobacco (1b), and environmental tobacco exposure (ETS) (1c) on provider-reported cancer-treatment toxicity (adverse events (both clinical and hematologic) and dose modifications). To determine the effects of tobacco on patient-reported physical symptoms and psychological symptoms. To examine quitting behaviors and behavioral counseling/support and cessation medication utilization. To explore the effect of tobacco use and exposure on treatment duration, relative dose intensity, and therapeutic benefit.