You're a new statistician in the field. What are the most important statistical analysis skills to master?

Don't underestimate the importance of exploratory data analysis! EDA is useful to help you find and understand the story behind your data. It isn't just one and done either. EDA could raise more questions than it answers and should be used in conjunction with data wrangling to create new features and start making sense of these new questions that arise.

One of the fundamental aspects of statistical inference is to state explicit assumptions about the statistical models that are being used for inference. Model uncertainty is often ignored in practice which could lead to erroneous conclusions. E.g., if you are using parametric models, you should clearly mention how you arrived at such a modeling framework and then provide some validations of the modelling assumptions using tests or hold out data. Further, it should be noted that even so-called non-parametric models are not free of assumptions (e.g., independence, autocorrelation decay rates), and other structural assumptions are often made and those need to be validated.

Algorithms are just like recipes, you need to choose the best as per your tastes! It is good to know which are the good and appropriate ones for your specific applications. No single algorithm is superior for all purposes. So, one needs to carefully weigh in various aspects before trying a whole bunch of them, just because these are available easily these days. The choice of metric (think of loss functions) is very crucial and its choice must be justified. The output of the algorithms heavily depends on the chosen metric (similar to choice of models in statistical learning). One should also try to quantify the uncertainty of the algorithms (e.g., by giving probabilistic error bounds) when used with data sets sampled from a population.

Eliciting the correct question from a client or subject matter experts is key. I have seen many teams work for weeks to come up with brilliant technical solutions that are never implemented because they answer the wrong question. Initial discussion where you build rapport with the client and extract their ideas is a great first step. It must be followed by one or more interactions in which the data scientist/statistician presents the proposed question back to the client to verify they really agree. During this process the statistician should make sure they hear the client and understand how the analysis will lead to a decision or action.

I used to be team R all the way, and then I learned Python... Getting really good at a statistical software can be useful, but don't let that be the only way that you learn to do an analysis. If you are lucky, you will get to use your favorite tech at your next job, but in all likelihood, you will probably have to integrate into using a new software that your new team is using at some point during your career. If you explore a little bit, you might just find that a task that you've been doing in your favorite software can be done so much easier in a different one!

I think design of experiments is largely taken for granted these days. But the data you want to model came from somewhere. It has an error/variance structure that can be very insightful when modeling.

One of the most important skills for a statistician (or really anyone) is to know how to be dumb. No one can know everything about statistical methods. We need to recognize our own limitations and ask others to educate us. A key skill is to find colleagues, friends and coworkers who have complementary skill sets and can advise you when you hit the limits of your knowledge. We then need to set aside our hubris and allow them to teach us.