Welcome to the Horse Forum.
To answer all of your question would require 2 or 3 books. So, I will just offer this one bit of advice:
Forget about barrels for at least 6 months or even a year.
Get your new horse very 'well trained' in all aspects of training. Start him like he was a reining horse and not a barrel horse.
Teach him to yield to every aid.
Get perfect control of all 5 body parts:
1) Head, (includes mouth and yielding to the bit and position of head carriage.)
2) Neck, (includes correct amount of flexion - always slightly bending in the direction of travel.)
3) Shoulders, (includes staying up between the rider's reins and legs and never pushing or drifting out and never dropping in. Shoulder control is probably the most important body part to have control of. A horse 'follows' his shoulders -- not necessarily his head.)
4) Ribs, (includes his entire body between his shoulders and his hips. A horse must be taught to bend his entire body if he is ever going to run around a barrel at speed.)
5) Hips! (This is the 'motor' -- and driving force behind speed and athleticism -- and you have to have complete control of this body part if you want it to propel him with the power and speed he needs to win. Go to any high level barrel Futurity or 1D barrel race. Watch the horse's hind ends at each barrel. Watch the horses that lose their inside lead on a barrel. They NEVER win. A horse does this because the rider / trainer did not teach the horse hip control well enough before they started running at speed. Once that lead is lost behind, the horse is only propelling himself with about 1/2 of the efficiency that a horse should have around that barrel and, particularly, leaving that barrel. Do not make that mistake.)
Without all of this done first, you will have a horse that you cannot control the parts of at speed. If you get complete control of all 5 body parts BEFORE you even look at a barrel, you will not end up with a barrel horse with major holes in his training.
If more barrel trainers trained this way, you would not see horses dropping a shoulder and knocking over barrels just like you would not see horses that 'fell out of lead behind' going around a barrel costing valuable tenths of a second each time. You would not see horses that wasted time fighting their rider's hands. You would not see horses that blow past a barrel or do not change leads at the right time.
There is no such thing as a horse that is too broke. EVERY single barrel racing rider (and their horse) that I have helped over the years has started their horse on a barrel pattern BEFORE they had it trained well enough to prevent problems. Then, once the horse has started running a pattern, the training holes showed up -- big time.
Most barrel horses that have been run at speed before they were 'broke enough' NEVER ran as good a pattern as fast as they would have if they had been trained well first.