strangely enough, based on empirical observations I have come to believe that the exact opposite may be true. It is the programming and computer science that is easier to learn on the go, and other sciences are far more difficult to acquire on your own.
I have met a large number good programmers that have never taken a programming or comp-sci class at a degree granting institution. Yet I have never met an accomplished biologists, chemists or physicists that would have acquired that skill without extensive formal training. I am not saying there aren't but it seems to be far less common.
I think programming requires more innate talent than other sciences. It is closer to art than it is to calculus. An analogy would be that with training anybody can learn how to solve a differential equation, but only select people can learn how to draw a pleasing image moreover those that do require far less training.
I think if you want to do bioinformatics (in academia or industry) then your plan is excellent. You have a good foundation in biology already. A bioinformatics masters will presumably introduce you to bioinformatics theory and applications. But, if you improve your CS skills on the side you will be a triple-threat. Although, a formal CS masters may not be necessary. I would definitely hire someone with your proposed CV.