This is the easiest to do on desktop! When you click on your username on the top right corner of the page next to the letter icon (inbox), a drop down menu should appear. One of the options says "Signature." Click that and it'll take you to a page where you can type a signature the same way you'd type a thread or a reply to a post!
Type the text you want to add, highlight the words you want to add a link to, and click the little chain link icon from the toolbar (when you hover over it, it should say "Insert link"). Then, you can go to your trip report in a new tab, copy the link, and paste it to the text. Click "insert" and your text should be linked! Don't forget to click "Save signature" when you're done adding your links.
I don't organize my TRs any other way, or print them out, but that is a really good idea -- it's probably smart to save them in some type of physical format in case something ever happens to the DIS.