As the subject states, is there a way to visualize the waves that the oscillators are producing, and the changes that things like the filter and envelop are making?
If you are on Artist or Pro take a look at SuperVision. It has a huge number of different meters & displays for examining audio.
Sadly just an Elements user at the moment. Any options for that? Third party?
There’s an abundance of third party metering plugins—a quick google search will reveal that.
Anything in particular you’re looking for?
Thanks for the direction. Part of being -very- new to all this is not knowing the names for things you’re trying to do!
Okay, metering plugins. That’s one avenue to explore. In particular I’m looking for a plugin that will show me the waveforms from the oscillators in a synth, and will reflect what changes the envelope and filter are having, to help me understand what’s going on when I move all these knobs and buttons 
Then perhaps you’re looking for an oscilloscope…?
I’m sure there are tons of videos and articles that can explain the basics of synthesizers. I would start by searching for something that explains subtractive synthesis first.
KVR Audio maintains a database of plug-ins. This is what I got when I searched for “meter” - you can further filter it for free or other criteria.
Also I bet there are some videos out there on how to use different types of meters.