My horses all knew how to run before I worked on their turns, but I think that's true with almost any horse.
I'm not sure I'll be much help but I'll give it a shot.
I'd start trotting the barrels. I'd start wit honly two and do a figure eight pattern, then build up to a lope and work on using your leg to drive her over while turning. Sit, lean, steer. Very important. Drive her as soon as she steps out of the turn.
I like to start practice by loping ten feet circles around the barrel and then tightening as we go, foot by foot inward. Make sure you tell the horse "Hey hey" when you're turning. Basic vocal command to dive in and turn sharp. Say "Ssh ssh" when you are starting to run and when the run past the barrel, without turning (For other gymkhana events, not so much in barrel racing.)
I like to have vocals commands as well, just to make my life a little easier.
But start slow and don't underestimate the power of circles. Lots of loping with the shoulder turned in, and work on using your legs. I rode a horse with no leg once, and I couldn't move her away from my leg so we knocked a barrel down.
Lots of circles. Lots of figure eights. Lots of leg.
That's basically what I do, and it hasn't failed me yet.
Pssh.I didn't pick up the wrong lead
It's called a counter canter...
...A very advanced maneuver.