OK, I went back and re-watched most of the first episode, because I felt like I was missing something. Here's what we know:
Red:
*Has a wife and daughter that he abandoned. Are they still alive?
*Is apparently "Obsessed" with Lizzie and even his associates are aware of this.
*Sent episode 1 baddy to Liz's house, specifically to get the husband.
Keen:
*Her father is a criminal who abandoned her.
*Her mother is dead
*The FBI seemed to be unaware of her father's criminal background, but for whatever reason, this hasn't been expanded on yet.
*Got the scar on her hand when she was 13/14, says her dad gave it to her.
- If she's referring to her dad who abandoned her, there's no way that Red's her dad, because she'd have been old enough to tell who he was
- If she's referring to her adopted father, there's something else altogether.
Now, maybe everyone but me was aware of this, but I know I missed the bit about her dad being a criminal as well.
Another minor thing, but it was bugging me ever since A/V Club pointed it out: Their big problem is that they can't understand how no one managed to grab a video of Red turning himself into the FBI, and can't buy the idea that the press/anyone has caught on to the fact that he's working with the government. When I rewatched this scene, they did shoot it where there was NO ONE around when he turned himself in, but FBI agents. But, I have to agree, it's still a pretty big plot hole.
And, I went back and re-watched the end of last week's episode and Red did refer to there being some adversary out there, so there is definitely a 3rd party waiting to be introduced.