Neural networks (NNs) have become an important tool for prediction and classification tasks in environmental science applications. Since many such tasks inform life-and-death decision or policy making, it is crucial to not only provide predictions but also gain an understanding of the uncertainty of these predictions. Until recently there were very few tools available to provide uncertainty quantification (UQ) estimates for NN predictions, but over the last two years the computer science field has developed numerous new methods for this purpose, and many research groups are exploring how to put these methods into practice for environmental science applications. In this work we provide a brief, accessible introduction to four of these UQ methods, then focus on tools for the next step, namely to answer the question: Once we obtain an uncertainty estimate (using any method), how do we know whether it is good? To answer this, we highlight four different evaluation methods that are particularly suitable to evaluate NN uncertainty estimates for environmental science applications. We demonstrate the UQ evaluation methods for two real-world problems: (1) estimating vertical profiles of atmospheric dewpoint (regression task) and (2) predicting convection over Taiwan based on Himawari-8 satellite imagery (classification task). We also provide accompanying Jupyter notebooks with Python code for implementing the uncertainty estimation and UQ evaluation methods discussed herein. This article provides the environmental-science community with the knowledge and tools to start incorporating the large number of emerging UQ methods into their research.