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Based on a recently developed variational method, we explore the properties of the Holstein polaron (HP) on an infinite lattice in D dimensions, where 1<= D<= 4. The computational method converges as a power law, so that highly accurate results can be achieved with modest resources. We present the most accurate ground state energy (with no small parameter) to date for polaron problems, 21 digits for HP in 1D at intermediate coupling. We show that the crossover to large effective mass of the higher-dimensional polaron is much sharper than the 1D case. The correlation length between the electron and phonons decreases significantly as the dimension increases. The eøtimes E Jahn-Teller polaron (JTP) is investigated in 3D and only quantitative differences are found between JTP and HP. In the strong-coupling perturbation regime, the mass of JTP is a factor of g√ π/2 smaller than HP where g is the …