Hi
Sounds like you are already an unofficial Super User. Basically the Super User is where Service Desk/Support Areas and Training get direct an inexperienced user for assistance. You become the go to person for your area and may be consulted if changes are going to be introduced to better understand business impacts
This is a good position to be in as a step up. Having access to a playpen system is even better
How do I tell what module we're running? I'm pretty sure it's MM, but there could be aspects of PP in there too.
It's not only module but integration points between the modules. As you step up, it's not a bad thing to be across a few modules. A trick is to search for the transaction codes in the role menu and see where it appears. It does sound like you have a mixture of PM (Plant Maintenance); PP (Production Planning); WM (Warehouse Management) and MM (Materials Management) - but I'm not a functional person so not best to comment
So, I need to take the bull by the horns and learn more about the system so I can present management with informed, intelligent solutions to our SAP woes.
In this case, don't focus on the module but start up at the business issue. You know what you have to do day to day on the job. Ask yourself what is the system lacking and then research from that point of view. You may find yourself jumping into different communities but if solving a business problem it's better to keep an open mind and not fixate on a specific module until you have the options. From here you can then learn more about the specifics and then prototype your solution in sandpit. Do this enough and you have stepped up from Super User to Functional person
As part of this step up (and one you find the specific area you want to excel in), ask management for training.
Good luck with it all
Cheers
Colleen