Well... you could help me in a project I am currently working on, to get some visibility.
In any case, research is a particular and complex world. The best strategy to do research is to get new ideas. To get new ideas, you don't wake up one morning with a new idea. You actually take ideas from others, and add a small bit. Newton said "if I've seen that far is because I was standing on the shoulders of giants". To do this, you actually have to know your "giants", before climbing them.
As a result, my suggestion is to read scientific papers of an interesting discipline you want to contribute on, possibly the same of a research group you have close access to. Don't worry if you don't grasp the paper. You probably will not get it immediately. You have to grow, understand. Perform a full bibliography search of the argument: look the citations, and get to know what's going on in the field relative to that problem, since the origins. Try to see open issues, interesting questions or problems in the current models you can contribute.
Try to build some kind of contact with a professor in the field or anyone under his supervision, by asking papers, for example. Don't bother him too much. They are very busy people. Go to a local university library to access the scientific papers, try to be present and visible somehow. When you have an idea, propose it to your favourite professor or postdoc (they are normally more interested in new projects, as their contracts are short and their stay depends on new things to do). If you get to have a chat, and get to have a project, you will probably end up with a completely different direction, idea or task than your initial one.
Continue this way. When an opening appears, you will be skilled enough to propose yourself and join the group without much hassle.
As you can see, it's mostly social networking, and it takes a lot of time and effort. I doubt you can do this while keeping your job. One fundamental recommendation: be careful of who you pick. Check how well funded s/he is, if s/he does active research, and how large is the group. Remember that postdocs and Ph.D. will be your main source of knowledge. Professors are too busy.