Ryuichi Sakamoto
While Ryuichi Sakamoto has a vast catalog of music, there are a select few really worth mentioning.
1. Neo Geo (1987) - A mix of Japanese traditional sound and western electronic pop sensibilities drives this album. A great starting point for those wanting to transition from Yellow Magic Orchestra to his post-YMO solo work.
2. Smoochy (1995) - This album fuses classical instrumentation (piano, strings, etc.) with electronic beats. A gem of an album, not a bad song in the bunch. This is the high-point in Sakamoto's solo career.
Note: Both 'Neo Geo' and 'Smoochy' seem to be out of print. If you want 'Neo Geo', you're going to probably have to buy a used copy. If you want 'Smoochy', you're probably going to want to purchase used, though buying import new is an option (though more expensive).
3. The Last Emperor (1988) - This easily accessible film score is the start of Sakamoto's collaboration with filmmaker Bernardo Bertolucci (he would later record the soundtracks to Bertolucci's 'Little Buddha' and 'The Sheltering Sky'). The soundtrack is not entirely Sakamoto, however. It's secondary composer is David Byrne (founder of new wave band 'Talking Heads') and his work for the soundtrack is also fantastic.
David Sylvian
Longime Sakamoto collaborator David Sylvian and member of 80's band Japan has only one really stellar album in his discography that stands out in my mind, 'Secrets of the Beehive'. His vocals are reminiscent of Bryan Ferry and the instrumentation is stellar; a very chill album. A beautifully coherent album, one track that stands out is 'Forbidden Colours' (a Sakamoto/Sylvian collaboration for the film 'Merry Christmas, Mr. Lawrence').
Harry Nilsson
One of the greatest pop musicians ever to exist, Nilsson created songs with great lyrical hooks. He wrote songs for The Monkees, penned the soundtrack to the musical film version of 'Popeye' (y'know, with Robin Williams), and was one of the first (if not the first) pop artist to ever release an album of standards. Three works of his I recommend whole-heartedly are 'The Point' (a beautiful fable), 'Newman By Nilsson' (a Randy Newman cover album; I know Newman's vocals are ridiculed, so are Dylan's, but both are excellent songwriters; Nilsson's vocals with Newman's great lyrics equals musical greatness) and 'A Touch Of Schmilsson In The Night' (his standards album), though his entire catalog is fantastic.
The Moffs
Obscure Australian indie band from the 80's, the Moffs have a great sound. Very laid back rock; I recommend their album 'Entomology', a compilation of singles. Every track is pure gold, though getting hold of their stuff is difficult. If you like the Stone Roses, you'll most likely dig the Moffs.
There is much more music I can recommend, but I feel that I have written enough for now.