Change the Player's Idle, Walking, Running, and Jumping Animation in-game?

I don’t know if I have to pull out the Animate script from the game and paste it into my game but change the animation ID or what

You can follow this guide Replacing Default Animation

Or you can do the other way as you mentioned of pulling the Animate script out of your character and placing it in StarterCharacterScripts and changing them there.

Because I tried that method and my avatar glitched.