An easy question at last!
12 Songs - great album of psychotic love songs, covering necrophilia, arson and stalking among other things. Buy it today.
Good Old Boys - a homage to the south that the south definitely didn't need, with not a wasted note on the album. Superb, superb, superb. Buy it at the same time as 12 Songs.
Sail Away - usually said to be his best, but I slightly prefer the other two, That doesn't stop this being a 5+ star album, that is never off my iPod. Best song about the slave trade ever written is on this one. OK, buy it at the same time as the other two, or you will regret it.
After that the going gets erratic. The first album Randy Newman is OK but over orchestrated and sounds more like a curio today. Little Criminals is a half decent album, with the best of it up with the earlier albums, and the worst of it somewhat lower. Similar is Trouble in Paradise, where his ongoing love affair with 70s west coast rock produces, well, slightly quirky 70s west coast rock. Not interesting enough to redeem it for me unluckily.
Later albums like Bad Love and Land of Dreams likewise have their moments (the opening twosome on Land of Dreams is Newman at his very best), but also lots of clumsy moments.
Finally the one to avoid like the plague is Born Again, which has nothing decent on it beyond its first track.