The distribution function for the first eigenvalue spacing in the Laguerre unitary ensemble of finite rank random matrices is found in terms of a Painlevé V system, and the solution of its associated linear isomonodromic system. In particular, it is characterized by the polynomial solutions to the isomonodromic equations which are also orthogonal with respect to a deformation of the Laguerre weight. In the scaling to the hard edge regime we find an analogous situation where a certain Painlevé III' system and its associated linear isomonodromic system characterize the scaled distribution. We undertake extensive analytical studies of this system and use this knowledge to accurately compute the distribution and its moments for various values of the parameter a. In particular, choosing a = ±1/2 allows the first eigenvalue spacing distribution for random real orthogonal matrices to be computed.