Indeed the question is broad and quite hard to answer I think. I'll give a try. I very welcome editing to improve this answer.
The field of bioinformatics is a big field. Bioinformaticians need basic knowledge in biology, molecular genetics, programming and statistics.
You may find courses on statistics applied to bioinformatics here (R-language) and here (I haven't watched these sources).
You seem to be mostly interested in is programming. I think that Python is a very good start to get in touch with programming. Programming might looks a bit scary when you don't really know what it is about but you can easily, in few days, acquire basic knowledge in this field and already solve some pretty neat problems. Many people actually have lots of fun learning how to program. And you'll probably be amazed by all the power this tool will offer to you. I personally really enjoyed learning programming in Python. I did it (I was mostly interested into object-oriented programming, you'll learn what it means) in a day or two with a very good source but unfortunately this source is not available in english. But there are tons of introductory documents, you'll have no difficulty to find a good one. I'd counsel you to directly download Python and to look at online courses on khan academy or EdX (I haven't watch them).
While Python is a much more powerful programming language than R, I think that, as a biologist, it is very important that you know about R. R is a programming language which is slow (compare to Python, C, Java, …) but it is very useful for statistical analysis and visual display of data. Also many people use R in bioinformatics (for phylogenetic analysis typically). I think that acquiring basic knowledge in R takes more time than in Python because we tend to use R because of its huge amount of already existing functions and therefore, we have to learn many of these functions before understanding that R can indeed be much more useful than Python for some tasks.
You also ask about the usefulness of programming. Well it is used in pretty much all areas of biology. It is used for analyzing empirical data, computer simulations in population genetics, graph theory, annotating DNA sequences, … I guess that 98% of biologists have at least some basic knowledge in programming. The main point about programming is that it performs calculation much faster than anything you could ever realized with your calculator. Typically, in bioinformatics, analysis of DNA sequences often asks for very intense calculation and asks for big computation power. Processes such as constructing phylogenetic trees, determining goodness of fit of evolutionary models, annotating DNA, aligning DNA sequences, analyzing microarray and many other things are all sorts of tasks that require programming.