So if you're a tester and you want that kind of breadth of knowledge, how do you start?
Pair with anyone who will let you.
Okay, maybe you're not in an agile environment and ya'll don't do pairing. In that case, the question is "can I sit with you for a while and...?". It's still pairing on some level, we just won't call it that! You get to learn about what they do and about the system from a different perspective, which can only help you test better. They get the benefit of fresh eyes, and of having to explain things to a newbie.
I don't think this was set up intentionally, but in every functional engineering team I've worked with, the test team is the glue that crosses silos. You want to be in that position. You want to be the team that sits with development and says, "You know, I heard the support team complaining about that taking a long time to do serially. I think they said they had a customer that added 200 widgets a day, and the customer was none to happy about having to do it one widget at a time." It may not get the workflow changed, but it starts the conversation, and you have a decent chance at fixing real problems before you ship.
So when in doubt, find someone and go pair. It's amazing what you might find, and you didn't even know you were testing!