What were their goals and how did they achieve them (or if they failed)? Were they living in poverty or were they already wealthy?

Their overall goal, initially, was to be the vanguard of a Socialist Revolution in Germany. They felt that if they attacked the state, the "hidden fascist" West German government would strike back with such overwhelming force against this relatively small challenge to the state that "the people" would then see their government for what it was (a fascist extension of the Nazi era) and would independently begin to attack the state themselves.

One could be snarky, or simply a realist, and go "how in the world could they ever have believed this would succeed?" But the group was buoyed by a belief that all capitalist governments were clearly doomed to collapse; they simply needed a vanguard group to help kickstart the Revolution and hasten the existing order's collapse. They were also encouraged by a public opinion survey in the spring of 1971 that indicated around 15 percent of Germans expressed a willingness to materially support the group, or would consider it. In retrospect this survey actually showed a level of support that was a mile wide but a millimeter thick. It was conducted really before the group had begun their deadly campaigns... and when the bombings began in may of 1972, that support eroded to virtually nothing almost immediately.

After the captured of the entire leadership of the group in May and June of 1972, the goals of the group (both the imprisoned leadership and the members still on the outside) morphed into essentially a single goal: secure the release of the imprisoned leadership. The 1975 German embassy takeover and bombing in Stockholm, the 1977 kidnapping (and later murder) of Hanns-Martin Schleyer... and other efforts during this era were all done with the specific goal of securing the release of the RAF leadership from prison.

After the failure of a joint Palestinian and RAF plane hijacking (German commandos stormed the plane in Mogadishu and rescued all the hostages; except for the pilot, who had been murdered earlier) the leadership in prison committed suicide in their prison cells in Stuttgart's Stammheim Prison.

So this led to a final morphing of their goals into more of a lets occasionally attack the state to remind people of the excesses of capitalism and German military partnership with the US. They no longer believed they were the vanguard of an immediate Revolution, no longer were trying to secure the release of their imprisoned fellow terrorists, but instead they were simply resigned to occasional bombings and attacks that would, in their minds, expose the flaws and hypocrisy of capitalist society.

Were they living in poverty or were they already wealthy?

Neither. The various members of the group were almost exclusively middle class. There were some members from fairly wealthy families, but most came from middle class upbringings. Of course one of the challenges of they group, in a sense, was that Marxist philosophy told them that it was the WORKING class that was going to rise up and fight for the revolution. Not the middle class. But inspired by the Frankfurt School teachings of Herbert Marcuse, and the writings of Frantz Fanon, among others, leftist students in the late 60s "learned" that they too were as oppressed as the working class, because they were enslaved by the trappings of capitalism like their TVs, their automatic dishwashers and the like.

Of course when the RAF finally did begin their war on Germany, counting on those millions of working class Germans to rise up and join the fight themselves, nothing like it happened. The RAF did not seem to understand that the working class of the mid 1800s as described by Marx bore little resemblance to the increasingly affluent working class of West Germany circa 1970. The "working class" of the 1960s and 1970s was not driven by their anger at their own "oppression" by those in charge. They were simply working, living their lives, and if they were politically motivated at all, they were as likely as not to be conservative as liberal.