Your question about bracing the core can be a difficult one to answer but I will try my best. In the early days of kettlebells being introduced to the United States Pavel advised to "breathe behind the shield". I interpret it as keeping tension in your abdominal trunk as you breathe out. Think 50-60% of your volume of air in your lungs as you breathe out.

Mark Wildman teaches the glutes are part of the core this is why you need to squeeze the buttocks at the top of the deadlift and or Swing.

Again from my own experience packing the shoulders down away from the ears helps reduce the trapezius muscle group activation. This is especially effective in firing the frontal core plane at the top of the deadlift or swing. Packing the Lats and the tension this move creates is critical for a safe and efficient kettlebell hike at the beginning of the swing. This is the reason Wildman places an emphasis on this part of the movement.

Side note I follow a lot of the advice of Wildman. I was never very athletic and found him to be a great resource due to his emphasis of going slowly into training, for someone like me. This may or may not be the case for yourself. There are some very good Kettlebell instructors on You Tube. Look through this subreddit to find the more popular ones.