Season two opens where season one ended, with Max in Manticore's (and Madame X's) hands, and most everyone presuming Max dead.
Logan, via Eyes Only, is putting the pressure on Manticore, calling attention to the Wyoming facility where Max is being held (without him knowing she's alive).
Max, meanwhile, has an escape plan, pretty much straight out of Escape From Alcatraz -- she has broken the mortar out of the cinderblocks of her cell and she has begun recon starting from the space behind that wall, which leads her into the basement of Manticore, and the anomalies kept there (and alluded to the first season). These would be the genetic misfits and experiments which produced humanoid but very certainly non-human creatures with different mixes of animal genes in their cocktails. There, she meets Joshua, a man with much more dog than she has cat, and he helps her to escape. He was one of the first creations, and he's been down there and free for a fair length of time. Unlike all the others, he apparently knew his creator, the genius behind the gene technology which made Manticore possible -- a man he refers to as Sandman.
While all this has been happening, Max has also met and rejected Alex (a clone of last season's Ben), who has been assigned to her as a "mating partner" -- as a result of Max's destruction of the gene bank at the end of last season, apparently that's the only way for Manticore to produce new soldiers. Being very much out for himself, he doesn't report her rejection, but instead treats it as a matter of course.
Max does, indeed, escape, with Joshua's and Alex's help. Manticore, however, has expected this possibility, and set her up to inadvertently kill Logan by using a tailored retrovirus that will only kill him, if he and Max engage in any intimate activity -- like kissing.
This revelation, given by Alex, sends Max back to the Manticore facilty to get the antidote, but upon arrival, she finds that the risk of exposure has caused the government to decide it was time to cut its losses, and to burn the facility to the ground -- with all the X's and Anomalies inside it.
Max sets them all free, and then flees herself, after Madame X is killed, but not before Madame X tells Max that she is, indeed, the genetic specimen they had all been looking for, and that she should seek out Sandman.
If all this seems a little bit less down-to-earth than most of the plots of season one, you'd be pretty accurate in your analysis.
Starting with season two, the plots and the events involved do, indeed, begin to become far more SF oriented and less human-oriented than before. I think this actually detracts from the series, but not a great deal. The writing is still top notch, and the plots, while slightly more improbable, are still as good as SF gets. I say this, mind you, as a life-long SF fan. I just think the human-style stories had an almost perfect balance between the underlying SF element of the show and plain old good storytelling about people. I still rate the show as a 10, just a softer 10.