What would you call "advanced" statistics? But let's start listing classes:

1) Intro to Discrete and Continuous Probability -- you'll need this for every possible path

Now we need to start branching out. Choose your adventure: applied or theoretical? Frequentist, Bayesian, Likelihoodist, or "Machine" Learning?

Your normal university statistics sequence will probably give you Intro to Frequentist Statistics 1 at this point. That's a fine way to go, but it's not the only way. In fact, many departments in the empirical sciences will teach Data Analysis classes, or the like, which introduce applied statistics before teaching you the theory, which would mean you've actually dealt with real data before you learn the theory. I think that might be a Very Good Idea.

Now let's hope you've taken one of the following paths:

• Data Analysis and Intro to Frequentist Stats 1
• Intro to Bayesian Statistics 1
• Intro to Machine Learning (with laboratory exercises to get experience)

From there I would recommend knowing linear algebra decently well before moving on. Then you can start taking courses/reading textbooks in more advanced/theoretical machine learning, computational Bayesian methods, multidimensional frequentist statistics, causal analysis, or just more and more applied data analysis. You should probably check what sort of statistical methods are favored "in the field" that you actually care about.