Hmm... this is ultimately your decision. I would advise to make/act in a couple films first to even see if you have a passion for making/acting in future projects.
And yes, it is every director and actor's dream to get big. It really is. Some people for the fame, some people just because they want to benefit the acting/directing community. BUT remember, I would advise for you to practice practice practice. Though I am still an amateur, I do have experience. And nothing makes your film/acting capabilities better than practicing. Sure you may think you're good enough... you've been the lead in school plays. BUT still practice.
So if you want to get involved with a web series, either look around for an open one and audition for it! If there aren't any... why not get a bunch of friends and make it yourselves? So remember, don't worry about getting into Hollywood. Worry about getting better and better. Then worry about getting famous.
Good luck and have fun!