We associate to any graph a sequence of integers called the gonality sequence of the graph, consisting of the minimum degrees of divisors of increasing rank on the graph. This is a tropical analogue of the gonality sequence of an algebraic curve. We study gonality sequences for graphs of low genus, proving that for genus up to 5, the gonality sequence is determined by the genus and the first gonality. We then prove that any reasonable pair of the first two gonalities is achieved by some graph. We also develop a modified version of Dhar's burning algorithm more suited for studying higher gonalities.