That was a problem for me early on too. After awhile you begin to hear the bass in every song you listen to. Sometimes I find myself not even listening to songs as a whole because I'm so concentrated on the bass. At first people don't
seem to hear the bass because it blends everything together, it just takes time and effort to be able to dissect everything in a songs. Keep with it and you'll thank yourself later on.
As for learning songs: You have the right idea IMO trying to learn Weezer because for the most part it's very simple and the rhythms are pretty straight forward (from what I remember), anyway. Since you are so new I would even go as far as to suggest tabs (power tabs are great for beginners) so that you start to understand what the bass is doing. Practice learning the songs slowly while doing everything correctly and in no time you'll be in full speed. You'll also probably want to consider heading down to the music store and buying a couple books on bass technique. If you start out doing everything right you will advance very rapidly and the pay off will be well worth your efforts. Anything I can do to help, let me know.
