I think to become serious about the bass you have to become a serious musician in general.
This starts with the way you approach your bass playing, in particular the way you practice. First of all, you DO practice on a regular basis; and WHEN you practice, you focus on fundamentals as much as on advanced stuff. This means you work on tone, timing, intonation, hand strength, scales, ear training, theory, etc.... stuff that's not nearly as sexy as popping and slapping but will make you a better player many times over than the local guitar store wanker.
But then it goes beyond just the way YOU play YOUR instrument and begins to touch on how you play with other people and how you approach music in general. Being a serious musician means that you actively seek out opportunities to play with other folks, whether it's in church, at school, after school, for the local top 40 cover band, whatever. You will learn at least as much about how to play bass by playing with other people as you will by practicing alone. In fact, practicing alone and playing with others go hand in hand, and one without the other is ineffective.
And even beyond that, it goes a lot deeper than just how well you play the bass with other people. Being a good musician also entails a lot of things about your professionalism. humility, and attitude. You have to show up early for practices; get your instrument in tune and keep it there; don't screw around during rehearsal when you're trying to get things done; help other players play better rather than criticize them; listen to the rest of the band when you play; and so on. Being serious about playing bass in part means you have to get yourself out of the center of the universe and learn to play as part of a larger group in ensemble.
And I think you have to make music a regular part of your everyday life, and begin to acquire a taste for all kinds of music. My band plays funk/jazz/rock but I love to listen to pre-15th century vocal polyphony (Anonymous 4 type stuff), Bach's choral works, and bluegrass -- stuff I'll never play bass with, but I learn so much about being a good musician by listening to it and it's beautiful music. If you want to play punk, then fine -- learn the genre and learn it well -- but don't shut yourself out of other genres.
There is actually a book called "Serious Electric Bass" by Joel Di Bartolo that is a great guide for getting serious... and you won't find a word about popping or slapping in the whole thing. Check it out.