I would like the user to be able to use hyperlinks within Word or Excel documents and redirect them to the correct document within the site collection, into the same or another document library.

What is the best way to do it?

If the document is moved to another location, the hyperlinks will be automatically updated? if not, is it possible to update them automatically?

Hyperlinks inside a document are not updated automatically. There is no OOB functionality available for it as of now. You might want to use 3rd party tools for updating hyperlinks.

Thank you, I assumed that there is not out-of-the-box, so my assumptions were valid. Whhich 3rd party solutions exist to help me on this task? Any reliable and user friendly solution to suggest?

We are not using any so I wont be able to recommend one. But a google search for this might prove useful to you. There are plenty of too