Well, I'm no expert. But from what I've seen, the most common needs have been linear/mulitvariate/logistic regression. I don't think anyone expects you to be a data scientist, but the line between analyst and data scientist is fuzzy. In general, I try to be able to do a little of everything, and talk to specialists as needed. So I can build a somewhat basic model in R to predict something, or say if one variable is correlated to another, but I run anything more complex by a data scientist.
One HW assignment I received gave me a large dataset, had me clean and dig into it a bit, prep some data for analysis, and build a model in R. Based on my predictions, what action would I recommend? What variables were significant? It was an interesting problem.
Maybe doing a course like this or this will help. Or if there are people with the skills you're looking for internally, approach them about doing a mentorship project. We have a process where you make a pitch to someone for a 3 month mentorship. Has to have clear goals, and a definitive end date. We've seen good success with that program.