2000 Honda Civic won't start after period of non-use

I've had issues recently with my car refusing to start, especially after periods of not driving it for a while. If I don't drive it for a couple of days, when I go to start the ignition, I hear a clicking sound, and I get the coughing sound of the engine turning over, but it won't start. (Sometimes it has more energy than others, and it will turn over faster and more aggressively, and other times I hear the clicking sound and that's it).

The battery is new, and it's clear each time that the battery is not drained of power (everything on the dash immediately comes on when you turn the key - indicator lights, radio, interior lights work).

In the past when this happened, I cleaned the corrosion from around the battery terminals, and the car started, reluctantly however. Today I got back from being out of town for four days, tried to start my car, and of course it wouldn't start. So I jump-started it, and it started fine. Then, after driving it for a few minutes, I came back to the house, turned off the ignition, and started it right back up, no problem at all. Two hours later I go back out to the driveway to see if it will start, and it starts up again with no effort.

The fact that it starts up normally after it's been running for a bit is throwing me for a loop. I was thinking maybe I should try unattaching the cables that connect to the battery and cleaning the end clamps that wrap around the terminals (but there doesn't appear to be much corrosion at all on the outside of the clamps, and right now I don't have the right wrench to remove the clamps). Does it sound like it may be the cables that connect to the battery? Any ideas?

Sorry for the long post, but I'm a complete dummy when it comes to talking about cars as you can probably tell, so any help would be greatly appreciated.