Welcome to the forum, Gary! Sounds like you had a very nice Christmas.
The HTR-6140 has (according to the owner's manual) HDMI 1.2a. 1.3 cables will work perfectly fine. You don't need Monster brand cables, for sure. You can get cables for about $10 to $15 (if you are running them about six feet) that will work just as well. Monoprice.com is a nice place to shop, but there are other stores, too. If you already have Monster brand HDMI cables, then don't buy new ones!
They'll work just fine - they are just more expensive than other cables that work just as well.
You would connect the DVD player to the receiver using an HDMI cable (HDMI output on the DVD player to an HDMI input on the receiver). That connection will carry the audio and video. You would play the audio through the receiver to speakers connected to it. You would also connect the HDMI output on the receiver to the HDMI input on the TV using an HDMI cable. This will pass the video from the DVD player, through the receiver, and onto the TV.
The subwoofer would be connected to the receiver. The receiver has a subwoofer output. Now, this begs two questions: (1) which subwoofer do you have, if any, and (2) which speakers do you have, if any? How you connect them will depend on what connections are available, and we can check that if you tell us what you have or are thinking of getting.
Adam