By watching all of these videos and practicing a bit on my own, do you believe I will gain the appropriate knowledge base to create a solid web application?
I do not see why not if you are motivated enough. I would even pursue 'social coding' on Github. Seek a project out and see if you can contribute? Learn from others after you have the basics down so you can ask constructive questions and hone some skills. Could serve as a mentor of sorts that you are seeking.