There can be a long list but to mention a few: (in no specific order)
P-value is NOT probability. Specifically, it is not the probability of committing Type I error. Similarly, CIs have no probabilistic interpretation for the given data. They are applicable for repeated experiments.
Problem related to variance dominate bias most the time in practice, so a biased estimate with small variance is better than an unbiased estimate with large variance (most of the time).
Model fitting is an iterative process. Before analyzing the data understand the source of data and possible models that fit or dontdon't fit the description. Also, try model any design issues in your model.
Use the visualization tools, look at the data (for possible abnormalities, obvious trends etc etc to understand the data) before analyzing it. Use the visualization methods (if possible) to see how the model fits to that data.
Last but not the least, use statistical software for what they are made for (to make your task of computation easier), they are not a substitute for human thinking.