This paper analyzes the performance of multicell massive multiple-input and multiple-output (MIMO) systems with variable-resolution analog-to-digital converters (ADCs). In such an architecture, each ADC uses arbitrary quantization resolution to save power and hardware cost. Along this direction, we first introduce a quantization-aware channel estimator based on additive quantization noise model (AQNM) and linear minimum mean-squared error (LMMSE) estimate theory. Afterwards, by leveraging on the estimated channel state information (CSI), we derive the asymptotic expressions of achievable uplink spectral efficiency (SE) over spatially correlated Rayleigh fading channels for maximal ratio combining (MRC), quantization-aware multicell minimum mean-squared error (QA-M-MMSE) combining, and quantization-aware single-cell MMSE (QA-S-MMSE) combining, respectively. During the derivations, we consider the effect of quantization errors and resort to random matrix theory to achieve the asymptotic results. Finally, simulation results demonstrate that our theoretical analyses are correct and that the proposed quantization-aware estimator and combiners are more beneficial than the quantization-unaware counterparts. Besides, based on a generic power consumption model, it is shown that low-resolution ADCs can obtain the best tradeoff between SE and energy efficiency (EE) under multicell scenarios.