In this paper, we propose a learning-based detection framework for uplink massive multiple-input and multiple-output (MIMO) systems with one-bit analog-to-digital converters. The learning-based detection only requires counting the occurrences of the quantized outputs of -1 and +1 for estimating a likelihood probability at each antenna. Accordingly, the key advantage of this approach is to perform maximum likelihood detection without explicit channel estimation which has been one of the primary challenges of one-bit quantized systems. However, due to the quasi-deterministic reception in the high signal-to-noise ratio (SNR) regime, one-bit observations in the high SNR regime are biased to either +1 or -1, and thus, the learning requires excessive training to estimate the small likelihood probabilities. To address this drawback, we propose a dither-and-learning technique to estimate likelihood functions from dithered signals. First, we add a dithering signal to artificially decrease the SNR and then infer the likelihood function from the quantized dithered signals by using an SNR estimate derived from a deep neural network-based estimator which is trained offline. We extend our technique by developing an adaptive dither-and-learning method that updates the dithering power according to the patterns observed in the quantized dithered signals. The proposed framework is also applied to channel-coded MIMO systems by computing a bit-wise and user-wise log-likelihood ratio from the refined likelihood probabilities. Simulation results validate the performance of the proposed methods in both uncoded and coded systems.