flash animation?

There is a program called Flash (I have version 5) and you can make movies like those found at newgrounds.com, or you can make games also found there. Whatever you do, don't plan on having long cinema's and then posting them on the websites, unless they're zipped (and then they still might be huge files), because no one wants to wait for 10 hours of downloading time to play a game that doesn't meet people's expectations.

As for the learning it part, in the Flash program, under the help menu, there is an option called samples and one called tutorials. The tutorials shows you how the stuff is done, and the samples are some small, basic, yet appealing items that can be done. For the complicated stuff you can't figure out, there are plenty of books out there.

Now, about bringing it into your game, look under the event commands, there should be one that says "PLAY MOVIE" and then it lets you load your movie file. Then you could divide the cinema between in-game sections and actual animated sections.
EXAMPLE: [show movie of bad guys attack town] SPRITES:hero moves towards goblin, and attacks, (goblin flashes, and disappears) [Show movie of Hero standing over goblin. Then runs to save others] SPRITES:hero runs to farmer and gets there too late. But then he realizes that his parents are under attack, so he runs over to them.[play movie showing hero running to his parents, but being stopped by Man in Large Cloak. Hero is struck down, and watches helplessly while parents are killed. Then everyone leaves, expecting hero to die, but he lives, and now has a quest]SPRITES:hero enters own house and gathers a couple things, then goes to stores and takes a few weapons, and starts his journey. GAME START! Or something similar.

On a final note, quality is of the utmost importance. You can't throw somthing together in a matter of seconds and expect it to be the greatest thing in the world. If you really want to see what flash can do, take a look at Mucha Lucha! And for special effects (particles, explosions, and others) you may need a different program.