Complex biological processes involve collective behaviors of living entities over different length and time scales. Angiogenesis, namely the creation and growth of new blood vessels from a preexisting vasculature, exemplifies this scenario and can be described by agent-based, discrete models that retain essential physical features of the underlying mechanisms. Here, we present a two-dimensional stochastic model of early-stage angiogenesis that tracks the migration and proliferation of endothelial tip cells. The latter are issued as a response to the emission of a vessel growth factor from a hypoxic region of tissue. Tip cells (modeled as point particles) move by chemotaxis, branch out, and coalesce. Their trajectories form the evolving vessel network. We derived a kinetic equation for the density of active tip cells, which is an integrodifferential, Fokker-Planck-type equation containing a source term for tip branching and a sink term with memory characterizing vessel merging. It is coupled to a reaction-diffusion equation for the growth factor concentration. This system is numerically solved by a convergent scheme with appropriate boundary conditions. Although quite expensive, numerical simulations of the kinetic model quantitatively agree with the ensemble averages of the stochastic model, which is computationally less costly. Theoretical studies of the kinetic equation help understanding the angiogenesis at mesoscopic scales. Indeed, in appropriate limits and simple geometries, the angiogenic process is driven by a one-dimensional soliton-like wave for the active tip cell density. The soliton moves from a primary vessel to the source of growth factor. The soliton location, velocity, and shape change slowly over time according to a system of collective coordinate equations, whose numerical solutions allow to predict the evolution of the angiogenic network at intermediate stages.