We present new GPU implementations of the tensor contractions arising from basis-related computations for high-order finite element methods. We consider both tensor and non-tensor bases. In the case of tensor bases, we introduce new kernels based on a series of fused device-level matrix multiplications (GEMMs), specifically designed to utilize the fast memory of the GPU. For non-tensor bases, we develop a tuned framework for choosing standard batch-BLAS GEMMs that will maximize performance across groups of elements. The implementations are included in a backend of the libCEED library. We present benchmark results for the diffusion and mass operators using libCEED integration through the MFEM finite element library and compare to those of the previously best-performing GPU backends for stand-alone basis computations. In tensor cases, we see improvements of up to 10-30% for some cases, particularly for higher basis orders. For the non-tensor tests, the new batch-GEMM implementation is twice as fast as what was previously available for basis function order greater than five and greater than approximately 10^5 degrees of freedom in the mesh; up to ten times speedup is seen for eighth-order basis functions.