When you start the pattern I recommend going both ways through the pattern and feel which way is better when you go. Next when you walk through stop your horse in the pocket then wait a few seconds then continue around the barrel to the next one. When you go through at a trot slow to a walk in the pocket and turn at a walk then trot to the next barrel. This is a great way to start your horse on barrels and teach them about slowing down and rating. You can add whatever cue to want to help. With my horse I half halt and say woah right before the barrel and that tells him to rate. With the above training methods done correctly your horse should never blow past a barrel. Also if you look on YouTube Fallon Taylor has a bunch of great drills, one of my favorites is the one barrel drill. This helped my horse learn to turn properly and because he only has one eye helped me figure out what cues I needed to tell him when to turn. Just be consistent in what you do but don't add speed until your horse has mastered it at the previous gait. If you do that's when people run into lots of problems with wide turns and blowing past barrels.
Walk it on your horse 100 times but acuratly, using your legs and hands exactly was your would if you were galloping it. Then do the same at a trot, make sure the horse doesn't lean into the barrel too early as this will teach him to knock it over when you go faster, inside leg as you approach the barrel so you are a bit wide of it and then outside leg and inside hand to do a sharp turn. Practice your canter flying changes as you hanged to do one on course and also practice stopping your horse after the flags. It's not about how fast you can gallop, it's about how accurate you are and how fast you can be accurate at
As brilliantly stated before, it is always good to start off slow so you and the horse are able to understand the basic fundamentals of the sport and also to descensitize the horse if he or she is a little frightened by them.