In this paper, we consider multi-agent learning via online gradient descent in a class of games called-cocoercive games, a fairly broad class of games that admits many Nash equilibria and that properly includes unconstrained strongly monotone games. We characterize the finite-time lastiterate convergence rate for joint OGD learning on -cocoercive games; further, building on this result, we develop a fully adaptive OGD learning algorithm that does not require any knowledge of problem parameter (e.g. cocoercive constant ) and show, via a novel double-stopping time technique, that this adaptive algorithm achieves same finite-time last-iterate convergence rate as nonadaptive counterpart. Subsequently, we extend OGD learning to the noisy gradient feedback case and establish last-iterate convergence results-first qualitative almost sure convergence, then quantitative finite-time convergence rates- all under non-decreasing step-sizes. To our knowledge, we provide the first set of results that fill in several gaps of the existing multi-agent online learning literature, where three aspects-finite-time convergence rates, non-decreasing step-sizes, and fully adaptive algorithms have been unexplored before.