For warmup, I just stretch and maybe do some squats, sprawls or push ups. I'll also bring a jump rope too.
I prefer to wear wrestling shoes when I grapple. I have been known to switch up and go barefoot sometimes then with shoes other times. I have two reasons for wearing shoes. The first is for traction of course. Some of those mats can be real slippery. The second, and most important to me, is that they help me apply head scissors which is kind of a bread and butter move for me. The head scissor is one of my top 5 moves and gets many of my tap out wins. With the shoes on, I can squeeze REALLY hard and my feet don't slip off like when I'm barefoot.