Tough one,
In short, freezing action is extremely hard unless you pay for a very fast lens (f2.8 will cover pretty much any conditions), however, if you will mainly be panning and having plenty of motion etc in your shots then the lens is almost irrelevant as your technique will achieve 90% of what you're after.
So it depends what sort of shots you are after really, if you like motion then the Nikon 70-300 or Sigma 70-300 will be excellent and they don't cost too much (both around £100). If you want to freeze stuff and isolate the subject by blurring the background you probably want the 70-200 2.8 (£1k +) Only issue is the D40 without the af motor, so obviously you would need to make sure whichever of the 70-300 has AF-S or equivalent so autofocus works. I can't recommend them enough though (I had the Sigma), I only sold it as I managed to get an 80-200 2.8 AF-S extremely cheap second-hand.
Frozen shot, 130mm, ISO 200, 1/2500 at f2.8.
The 70-300 would probably give you a black image at that shutter speed in those conditions (under trees) but the 80-200 does it without breaking a sweat.
However go with some motion and there isn't too much in it...
Sigma 70-300mm f4-5.6. 1/320 @ f8.0
Nikon 80-200mm f2.8. 1/125 @f7.0
That's a £1000 difference in lens in the above 2 pics, but you would be hard pressed to see the difference unless you frequently blow your pics up pretty big. Obviously the 80-200 2.8 comes into its own in other situations, but the 70-300 will do the business as long as you play to its strengths.