Let me preface this by saying that I don't listen to much Weezer anymore. In fact, the only record of theirs I can stand to listen to is this 2001 self-titled one. (Side note - quite a few of their albums are self-titled and go by the color of the record sleeve, so keep that in mind when going through their discography, if you choose to do so.) Diehard fans will swear by their debut (again, self-titled but Blue) or their follow-up, Pinkerton, but I'll stick with keeping Green as the only Weezer record on my virtual shelf.
This is some of the best power pop/alt rock to come out of the early aughts. Ten songs over about 30 minutes - just the perfect length to draw you in but not make you want to turn the music off. In other words, it's all killer and no filler. The best song on the record is 'Island In The Sun' so I'd start out with that one. Then just circle back to the first 3 songs: 'Don't Let Go,' 'Photograph' and 'Hash Pipe.' By that point, though, you're already almost halfway through, so might as well continue listening to the end.
Finally, I'll leave you with this excellent Saturday Night Live skit illustrating the dichotomy between fans of early Weezer (up to and including Green) and everything that came after. I can totally relate to Leslie Jones' character!