When you do have a good foundation and you and your horse are ready to start with the barrels you should keep the pattern at a walk until he's used to it, then try it at a trot and see how he does. That's what I did with the TWH that I had and he loved to do barrel racing. You don't want to start out really fast though.
Hope you and your horse do good.
R.I.P Duke <3
No hour of life is wasted that is spent in the saddle.~Winston Churchill