I know the feeling. It gets worse the more plugins you have installed.
Here's the strategy I use with a new plugin:-
First check the mini web-page which appears in Plugins Manager. If you're lucky, the author will say where it's installed to.
Follow the link to the homepage (all plugins have this). Often this will just take you to github, but sometimes to a blog post or manual.
Check the tags. This is one of the ways authors have of showing which categories the plugin falls into. If you see the tag Vector, Database, Raster or Web, try that menu.
Right click the toolbar, and look in the context menu (this tends to be the best approach for newer plugins)
Check the Processing Window.
In very rare cases, plugins create their own menu (e.g. mmqgis)
If all else fails, I use @Kurt's suggestion and read the code.
I had a look to see if it's possible to do this in code, but it's not a trivial task... the job of registering the plugin to the gui is done by the plugin author, not plugin manager.
I've tried some static code analysis (scanning python code for the method calls to register menu items) but it's error-prone.