Mozart, the late concerti are essential (~no. 20 onward for piano, I'd say). He's probably my favorite concerto writer actually: he always took great care to make sure that the soloist and the orchestra are always perfectly balanced, and unlike some of the romantics, he took the form seriously enough to create works that go beyond their claims to virtuosity.
He's definitely not the kind of composer that's going to garner a lot of praise around here, but I think that his place amongst the great is well deserved: there's a perfection, an elegance, a purity, a clarity and a simplicity (only apparent, there's a wealth of nuances and intricacies beyond the notes in Mozart) to his music that's you don't find in other works. I think he was one of the composer that best understood the importance of saying more with less, and his later works show that this was something he was enormously talented at.
Mozart might have had comparatively lower aims than Bach, Beethoven and the like, but what he tried, he accomplished with such excellence that it really isn't anything you could hold against him. His music is the most formidable hymn to life that I could think of.
Haydn, the keyboard sonatas are brilliant and entirely different from the ones by his two famous contemporaries. If I were to describe them as a whole, I'd say that they're mostly very lively and driven, with a knack for expressing both the humorous and the tragic (often in the same movement). And of course, they're also full of the kind of "tricks" that Haydn is famous for: there's no such thing as being bored while listening to this man's music. I'm not familiar with much else of his output, but the few London Symphonies that I've heard were all every bit as fascinating as his sonatas.
Schubert, you didn't mention his Lieder! (ok, it's chamber music, but they're too important not to be mentionned on their own). Winterreise and Die Schöne Müllerin are probably the two most important cycles of the whole art song repertoire, and that's for very good reasons. They're completely and thoroughly unavoidable.
And on Tchaikovsky finally, he does stands on the wrong side of the romantic period, but don't let it get in the way of your appreciation of his magnificient Symphony No. 6. The finale's one of my highlight in the whole symphonic repertoire actually.
and finally finally, Scarlatti gets my vote too, 1685 was quite the year for western music.