So, I'm new to speedrunning and all I have is an xbox. How do other console players time and take splits?

Splitting and timing is generally done through computer programs like LiveSplit. To combine splits and gameplay into one video, you would need to route your gameplay first to your computer through a capture card. However, this equipment can get expensive, so most communities don't require splits in your video, and many will accept just using a phone to record your screen.