What is the best way to shrink HD Quicktime MOV files

There are a number video transcoders that you could use to covert mov files to different formats suitable for uploading. Handbrake is a popular one that has both a GUI and command-line version (handbrake-gtk or handbrake-cli). To use Handbreak on Ubuntu, you need to add a PPA:

https://edge.launchpad.net/~handbrake-ubuntu/+archive/ppa or the more current https://launchpad.net/~stebbins/+archive/ubuntu/handbrake-releases

I personally like, transmageddon which is a GUI app with nice simple pre-sets. It's available in the archives.

Another option is arista, another GUI transcoder available in the archives. In fact, it is a "Featured Application" in Ubuntu Software Center.

ffmpeg is another command line option that offers a lot of flexibility but can be a bit complex.

I use those options with ffmpeg (example with a video of 1m40s of 725MB 4096x2304):

MP4 conversion, Medium video quality with sound codec aac(245MB):

ffmpeg -i a.mov -f mp4 -vcodec libx264 -preset fast -profile:v main -acodec aac output.mp4

MP4 conversion, Medium video quality with sound codec mp2(218MB):

ffmpeg -i a.mov -f mp4 -vcodec libx264 -preset fast -profile:v main -acodec mp2 output.mp4

Avi conversion, Low quality (120MB)

ffmpeg -i a.mov b.avi

In addition you can reduce resolution with option

-vf scale=1024:576

MP4 conversion, Medium video quality with sound codec aac with rescaling (21MB):

ffmpeg -i a.mov -f mp4 -vcodec libx264 -preset fast -profile:v main -acodec aac -vf scale=1024:576 output.mp4