Well, Team Rocket's goal is harder to guess at then Aqua, Magma, and Galactic, but not much so. I shall tell you what I believe is correct on both generations.
Team Rocket Generation 1: Led by Giovanni, they functioned as the Mafia of the pokemon world. Stealing pokemon, attacking and even killing trainers, and selling drugs... not to mention that they're stealing money from people via the Rocket Casino. They want money, and they want to be feared.
That is, until we reach Silph Co. Team Rocket invaded with one thing in mind, the Master Ball. They were going to mass-produce the Master Ball, and capture powerful pokemon with them, such as the Legendary Birds, Mew, Mewtwo, Dragonite, Snorlax, Lapras, Gyarados, ect... allowing them to build an army to overthrow the Elite 4 and the Pokemon League, allowing them to rule over Kanto. But they bit off more they could chew, and Giovanni had to disban Team Rocket, but it wouldn't be the last time we would hear of them.
Team Rocket Generation 2: Led by Apollo, they began to resume activities in Johto, selling Illegal foods, practicing Pokemon Abuse multiple times, once again trying to steal money, to produce a signal that Giovanni would hear, and to find a signal to make their pokemon into powerful machines of war.
While the last one was perfected by Cipher, they layed the foundation down by forcing Magikarp to evolve early, in order to produce Gyarados, which would net them the perfect army of killing machines. But they didn't want to do anything too drastic until they could get Giovanni back, so they can allow him to take charge.
Team Rocket Anime: Led by Giovanni, this team seems no different then the game, but it seems more megalomaniacal(sp?) Then the games, so I put it in. Giovanni wants to rule the world, and he wants power. He is interested in Ash's Pikachu because it is way too powerful to be normal. He funded the creation of Mewtwo, and attempted to use it, but it fled. I don't remember too much about him, but he does show slight interest in expanding Rocket control, but not enough to persue it formally, sending three mid-class rockets to attempt to build a base.
Team Rocket Manga, and Neo Team Rocket: Don't know enough to make a guess, sorry.