If I have a file with many pages, what is the best/easiest way to select certain pages in that file and then save those pages into their own file?

You can

Select necessary pages in the Pages List

and click

File | Save pages as

You will see “Save selected pages” dialog

Choose the necessary item and click OK. Each page will be saved in a separate single-page file or Selected pages will be saved in one multi-page file.

Available programs with this feature:

• Advanced TIFF Editor Plus (free download, home page);
• Advanced TIFF Editor (free download, home page);