The show was Battles BC which aired on the history channel years ago and I found the rivalry between the Egyptians and Hittites fascinating, but would love to learn more about the ensuing conflict between the three of them. I did try some searching online but I’m not exactly sure where to start or how to properly word what I’m looking for so I’m here to ask the experts.
Thank you for what you do on this subreddit!
I wrote a lengthy set of posts about Egyptian-Hittite foreign relations a while back, so that should give you an overview.
For more information on the 13th century BCE, I recommend Marc Van de Mieroop's The Eastern Mediterranean in the Age of Ramesses II and chapters 7-10 of his A History of the Ancient Near East. Also check out Beyond Babylon: Art, Trade, and Diplomacy in the Second Millennium B.C., available for free courtesy of the Met Museum.
Letters of the Great Kings of the Ancient Near East by Trevor Bryce and Brotherhood of Kings: How International Relations Shaped the Ancient Near East by Amanda Podany are excellent introductions to ancient diplomacy in general, though the latter ends with the 14th century BCE and does not cover Egyptian-Hittite relations.
Finally, Warriors of Anatolia: A Concise History of the Hittites by Trevor Bryce is a good general introduction to Hittite history and society.