I think it's interesting that the two Russian composers you listed are both so closely associated with ballet. Stravinsky's collaboration with Sergei Diaghilev catapulted him to fame, and was a pivotal moment in the history of both music and dance. Stravinsky reinvented his musical style numerous times, but the most famous example is his "Rite of Spring."
The story is that of an ancient tribal ritual, which culminates in the sacrifice of a virgin, who must dance herself to death to appease the natives' gods. Instead of wearing traditional dance costumes, the dancers were covered from head to toe in folk dress, Vaslav Nijinsky's choreography abandoned conventional dance postures (the dancers were knock-kneed and pigeon-toed), and the music itself was primal and grotesque, with heavily accented ostinatos and dissonant harmonies.
I would love to go back in time to see the premier--it nearly ended in a riot! Here I will quote from Wikipedia:
On the evening of the 29 May the theatre was packed: Gustav Linor reported, "Never ... has the hall been so full, or so resplendent; the stairways and the corridors were crowded with spectators eager to see and to hear"... The Rite followed; there is general agreement among eyewitnesses and commentators that the disturbances in the audience began during the Introduction, and grew into a crescendo when the curtain rose on the stamping dancers in "Augurs of Spring". Marie Rambert, who was working as an assistant to Nijinsky, recalled later that it was soon impossible to hear the music on the stage. In his autobiography, Stravinsky writes that the derisive laughter that greeted the first bars of the Introduction disgusted him, and that he left the auditorium to watch the rest of the performance from the stage wings. The demonstrations, he says, grew into "a terrific uproar" which, along with the on-stage noises, drowned out the voice of Nijinsky who was shouting the step numbers to the dancers. The journalist and photographer Carl Van Vechten recorded that the person behind him got carried away with excitement, and "began to beat rhythmically on top of my head", though Van Vechten failed to notice this at first, his own emotion being so great.
Monteux believed that the trouble began when the two factions in the audience began attacking each other, but their mutual anger was soon diverted towards the orchestra: "Everything available was tossed in our direction, but we continued to play on". Around forty of the worst offenders were ejected—possibly with the intervention of the police, although this is uncorroborated. Through all the disturbances the performance continued without interruption. Things grew noticeably quieter during Part II, and by some accounts Maria Piltz's rendering of the final "Sacrificial Dance" was watched in reasonable silence. At the end there were several curtain calls for the dancers, for Monteux and the orchestra, and for Stravinsky and Nijinsky before the evening's programme continued.
Both Disney's "Fantasia" and "Fantasia 2000" used Stravinsky's music, the "Rite of Spring" to depict the last of the dinosaurs who roamed the earth and their final destruction, and "Firebird," to depict a story of death and rebirth. Cartoons are fun, but it seems like most kids are somewhat ambivalent about the two Fantasias. What I really, really, really recommend, if you can do it...is to go to a live performance.
I have been lucky enough to perform the Firebird Suite a couple of times, and there is a moment that literally has the audience jumping out of their seats. You'll want to be in a good hall, with a good orchestra. If that is not possible now...wait until it is. OR...try to set up a performance in your living room with a really good sound system. I also was able to watch Ballet Idaho's dress rehearsal of "Rite of Spring" once, and it was very powerful.