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.
It depends on the flower! Some flowers are incompatible with themselves, so no fertilization will happen. Self pollination is not very common, but it does happen.
There are two types. Type I is when the pollen fertilizes a carpel in the same flower. Type II is when pollen fertilizes a different flower on the same plant. Self pollination helps for rapid growth of a population, but can decrease fitness, because all individuals in a population are genetically similar. If a pathogen moves through the population, it could kill all of the individuals, because none vary to have resistance.