Here's how I read the character intentions:
Tekkadan: Want to make a name for themselves as a mercenary group. Their current job is bodyguarding Kudelia and Makanai and getting them to the capital of Arbrau. They feel that they can't back out of this job because no one would take them seriously if they did, and they'd have to abandon the bulk of their Mobile Suit forces to do so. Further, while they are allied with Teiwaz currently, their friendship with that organization is dependent on results. If they cease to be useful and become a liability, Teiwaz will drop them. As for making an enemy of Gjallarhorn, well, that kind of happened right at the beginning, but Gjallarhorn's influence beyond Earth itself is tenuous at best, so Tekkadan can, if necessary, stay out of their way and make a comfortable living in Jupiter and Mars.
Kudelia: Wants to generally improve the lives of the impoverished of Mars (and now the lower class of the Earth Sphere as well). Current goal is to make a speech at Arbrau, one of the Earth-based powers that has shown some support for her cause.
Gjallarhorn: Wants to maintain the status quo, with themselves on top of the world. Tekkadan's been making a fool of them for a while, so they want the group gone. But they have to careful of political backlash since Tekkadan has a few friends in high places and are heroes of the people.
McGillis: Claims to want to reform Gjallarhorn. His true agenda may be something different, as he's doing things behind the back of his supposed allies. Full scope of his plans remains a mystery.
Gaelio: Seems to genuinely want to reform Gjallarhorn, but is being left in the dark regarding McGillis's full plans.
Ein: Wants to KILL ALL THE TEKKADAN!