You didn't miss anything; the gap between those two episodes needs to be filled by by interpolating based on the rest of "Day of the Moon". There hasn't been any attempt to fill in those details on-screen in any form.
Primarily based on The Doctor's conversations with Canton Delaware later on, we can put together the basic idea, which is that The Doctor and Canton Delaware hatched a plan to try to fight the Silence. After the end of "The Impossible Astronaut", they must have put together the following:
- There was a creature that could wipe memory and control minds running around the US.
- Amy's ability to take a cell phone photo of one of the creatures gave The Doctor's team an advantage that The Silence didn't plan for -- proof that the creatures exist.
- The Doctor realized that they needed to figure out how badly The Silence had infiltrated the US.
- The Doctor also knew that The Silence had to be kept in the dark about how much The Doctor knew, since all memory of them should have been erased.
So, the plan that he came up with was to have Delaware declare the four non-Americans to be spies (given that it was Nixon in office, not that much of a hard sell), and "chase" them all over the country. He would pretend to have captured The Doctor, and the other three would be fugitives. Once they had learned everything they could, Delaware would finally hunt them down and "kill" them. That way, the Silence would have no idea what the US government did or didn't know about the "occupation".
Because they can't remember seeing one of the creatures, The Doctor came up with the idea to mark their arms with counters every time they see one. While they're looking at a creature, they know to mark their arms up, but as soon as they turn away, the only evidence they have is that their arms have more marks. Note that this means it's likely that at least some of Canton's confusion over what's going on may even be real, but that's just conjecture.
We pick up "Day of the Moon" after their fact-finding missions are over, and Delaware has somehow been alerted to go pick them up. Note that he's always the one that shoots at Amy/Rory/River -- he isn't trying to kill them, merely knock them out so they can be transported back to The Doctor, where he's building a "cell" around the TARDIS.
As far as what happened to "the little girl", we know she somehow escaped the suit, and the Silence. The girl in the space suit is the same one seen at the end of "Day of the Moon" starting to regenerate, and suit is the same one The Doctor and River examine later on. Note that the hole in the visor of the suit is rather high up, meaning Amy probably missed; even if not, The Doctor points out that the suit has extra life support systems, so it likely protected the girl from any harm.
Unfortunately, because they encountered The Silence in the warehouse, all of their memories of those events are gone, so we never learn exactly what happened. We don't know why The Doctor let the girl escape, but remember that the Silence appeared to be following Rory back up from the basement, plus Canton was injured at the end of the episode. Most likely, there was a lot going on right after the shooting, with The Doctor struggling with Amy and/or the Silence showing up, and in the confusion the girl just escaped the damaged suit and fled.