moving Saved data

Nothing can be transferred between profiles.

You can either get a transfer cable like davemcfly85 suggested or if you don't want to spend the extra money, you can transfer upwards of 32GB of data on a USB Flash Drive.

The last time I changed console's over all I had to do was take my save games from one console using my USB Flash Drive and then just re-downloaded all of the content that I wanted to actually play.

If you'd like to take that route you can simply copy the files from your hard drive using a USB by following the directions
here.

If you want to re-download any of your content you can do so by going to your Xbox 360 signing into your account and then going to Settings > Account > Download History. Alternatively, you can also queue up downloads on the website by going
here, these downloads will start once you turn on your Xbox 360 and sign into your account.

Since you're getting a new console, you will also want to do a license transfer so all of your purchased content can work when online, offline, and for other profiles (if there are any) on the console. To learn how to do that, you can visit this site
here.

You should not normally need to worry about copying the game saves as I think they don't get deleted from the console when you uninstall the game and as long as your console is connected to Xbox live they will also be held there.

all you need is this:
support.xbox.com/.../set-up-transfer-cable

Thanks for you help, people. I wasn't aware the saves were uploaded to cloud storage.