With the parent file open, go to File > Pack and Go.
This will allow you to copy the file and all of its dependent files (Parts, sub-assemblies, etc.) to another location. Most of the options are self-explanatory, but I'll touch on a couple of them. If you use Toolbox there's a box near the top left you can de-select to avoid copying them. There's also an option near the bottom left to send the files to a Zip file, which is handy if you need to send them to someone (or post them in a forum).
Changing the names of the new files is a good policy to make sure you don't get unintended changes to your original files. There are three ways to do this:
1. Double-click on the file name in the "Save to Name" column. That will allow you to assign a new name to individual files. This works fine if there aren't too many files, but for those with quite a few use 2 or 3.
2. There are checkboxes near the bottom right corner where you can add a suffix or prefix to the new file names.
3. Use the "Select / Replace" button (near the center, just below the list of files) to replace text in file names (such as project numbers) with new text. This function can also be used to exclude some components (such as library parts) by selecting "In Folder" from the Search drop-down, entering a key word, and then selecting "Uncheck item(s)".
When you've finished and clicked "Save", be sure to close your original file, then open the new files to make your changes (I learned this the hard way).