A great way to start learning is by reading! I have a short list of books here, all about different areas of botany. [Followers, if you have any suggestions, please send them my way!] Getting outside and identifying plants can be a great way to learn. If you happen to live in the northeastern US, Newcomb’s Wildflower Guide is awesome, otherwise get a local field guide (your local library should have some) and go on hikes. Wikipedia is actually a great place to find plant information, so if you see a plant or plant family, on here or anywhere else, it’s worth googling to learn more!
That said, my experience with botany degrees is that they do teach you the basics. I was required to take entry-level biology in my freshman year, as well as a basic ecology course my sophomore year.