<p>My goals are focused on software application development, and maybe web application development, but most likely desktop applications. I'm embarking on a path to becoming more familiar with C/C++, but should I go much lower than that, into assembly? Or would I not have a benefit for my long-term goals?</p>

I wouldn't start learning ASM. If you want to learn C/C++ then start with that. As the quality of your code matures, you may find you have a need for ASM. 99% of the time you won't, but every now and then you might need it.
Also, it does help to know ASM in terms of understanding what C/C++ is doing behind the scenes. But again, until you get more advanced, you probably won't have a need for it.