So, when you're looking for the date range, you can find the "Analysis Date Range", but then how do you decide what is the date coming after that?

I would say that you're looking for:

1. First, the string "Analysis Date Range : " with an arbitrary amount of whitespace around the ':' (/Analysis Date Range\s+:\s+/)
2. Then, at least one digit, which we want to remember (/([0-9]+)/)
3. Then, a slash (which needs a backslash to escape it) followed by at least one digit, and record the digits but not the slash (/\/([0-9]+)/)
4. Then, another slash followed by at least one digit, and again record the digits not the slash (/\/([0-9]+)/)
5. And the rest doesn't matter

Combining that all into one:

# i modifer to make it case insensitive, just in case.
if ($line =~ /Analysis Date Range\s*:\s*([0-9]+)\/([0-9]+)\/([0-9])+/i
+)
{
  print "I found an Analysis date with month/day $1, $2 in year $3\n";
}else{
  print "This line did not have an analysis date that I recognize in i
+t\n";
}
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Another way to go, would be to grab larger chunks and then split them later:

1. "Analysis Date Range : "
3. Then a pile of digits-and-slashes (start date) ([0-9\/]+)
4. Then whitespace and digits-and-colons (start time) \s([0-9:]+)
5. Then more whitespace and some digits-and-slashes (end date)
6. Finally, whitespace then digits-and-colons (end time)

PS: Is that date string "month day year" or "day month year"?