Falling Skies tells the story of the aftermath of a global invasion by extraterrestrials. Is the date of the invasion ever stated, or could it be inferred from other clues?
Based on the information from the pilot, the time frame appears to be right around six months prior to the start of the series. We learn that this is how long Tom has been with the Second Mass at the start of the first episode. He joined the army pretty soon after the invasion, when his wife was killed and Ben was kidnapped.
However, even knowing the time frame doesn't give us an exact date, because we don't have a clear timeline of what "day" it is, or even how much time has elapsed since the first episode. That kind of record keeping has been mostly eliminated in the show. It appears to be spring/summer in every episode we've seen (if they are going from Boston to NC in the winter it would definitely be cold, possibly even snowing, and the trees should all be dead). The also haven't celebrated any major holidays along the way, most of which occur in the October-April time frame.
So, the best we can do now is make some educated guesses.
We don't know how long the initial invasion itself lasted, so it's possible that there was a period of days or weeks between the first attack and the point that Tom joined the military. However, the show gives a very strong impression that the aliens destroyed civilization in a very short time frame, so I think that "six months" is a pretty accurate estimate.
Since we know that three months passed between S1 and S2, the most likely explanation is that the show started sometime in late spring/early summer, that S1 lasted perhaps 4 weeks of in-universe time, and S2 started in the fall of that same year. (Possibly, in season three we might see Thanksgiving/Christmas celebrations, which would help pin down the timeline.) The only part I'm completely unclear on is the amount of time they actually spent in the high school in the second part of S1; I'm guessing not more than a month but I could be way off.
If we use the actual air date of "Live and Learn" as a basis, the invasion would have been sometime in January, 2011, which actually fits the proposed timeline pretty well. Season One would have run over the course of June-July 2011, then a 3-month jump, and Season Two would begin in October.
I believe that Weaver's original plan in S2 was to head to the Catskill Mountains, which are in SE New York, so he was very likely still in the Boston area at the time. A journey from the Boston, MA to Charleston, SC is ~1000 miles. On foot, an army could probably cover ~20 miles in a day; with motorized equipment to carry supplies, wounded, etc. they would move much faster, but I don't have a good feel for how much. (I feel like Weaver has, at least once, mentioned how much ground they can cover in a day but I can't remember it.) If we use a conservative estimate of twice as much, that still means the trip takes only 3 weeks, putting us into November. (This has the convenient effect that the weather would stay mild longer as they went further south, so you wouldn't notice the lack of scenery changes.)
This is the best estimate I can really come up with for a timeline; hopefully we get more clues next season, but I think the most we can really say with any surety is that it's probably been "less than a year" since the initial invasion.
The 'date' is never stated as far as I know, but I think that's the point. It's meant to be Present Day since there is no fictional human technology. Even mobile/smart phones were not in play since the tech wouldn't work with all the infrastructure decimated. If the answer to your other question is found, along with the date of birth of any character who had a birthday in the story, then you could calculate it. (I think Mason's youngest son or Mason or Weaver had a birthday in one of the episodes, can't recall who.)