I think you are pretty much spot on yourself: besides formally studying bioinformatics, you want to somehow get to do actual bioinformatics projects.
Whereas practical exercises on courses can be great for efficiently illustrating how to do specific types of analyses, they are typically streamlined to avoid many of the practical problems that we all have to deal with every day (format conversion, identifier mapping, malformed input files, installing tools, parallelizing jobs on a cluster with a queueing system etc.). To gain the experience to deal with these many issues efficiently, there is in my opinion no substitute for doing real-world projects, where things have not been "pre-digested" to make your life easy (like it has in exercises).
Where you do such projects is less important than that you make them. If you are very fortunate, you can land a part-time student job in R&D in industry like I did. That both gives you the aforementioned experience, helps you build a professional network outside your university, and obviously puts some money on your bank account (which is obviously wonderful as a student). You can also see if there are any ways to do small individual projects at your university and get course credits for it - many universities have some sorts of "special courses" that basically just require that a professor is willing to supervise it and run an oral exam at the end.
Long story short, you would be wise to seek out real projects any way you can. It will give you experience and a professional network, which can also serve as references when you apply for jobs later on. If it also gives you some money on the side, that's wonderful, but it is not what really matters in the long run.