Welcome to the world of horses! Watch out for barrel racing though .... it's like an addiction problem, LOL.
Need for speed!
It won't be so much of a time frame to know when you will be ready, but how you are progressing with your training as a rider.
Check out the barrel racing sticky we have: Barrel Racing Exercises and Drills.
You will need to be comfortable riding your horse at all speeds: walk, jog, extended trot, lope, and fast gallop.
You will need to have a good seat and good legs. Your legs are very important to cue your horse to do specific things with its body:
--move the hindquarters
--move the ribcage
--move the shoulders
These things need to be done with your body language, weight, and legs. NOT with the reins.
You will need to have soft hands. That means you can't be hanging on the reins for balance (hence why you need to have good legs and body cues), and you can't be hard on your horse's mouth when running the pattern. Use your legs, legs, legs!
Barrel racing itself requires a lot of concentration, so your riding needs to be muscle memory. You don't have time to think about staying balanced AND turn a barrel. Your balance needs to be automatic.
And whatever horse you decide to use for barrel racing, your horse needs to be able to:
--walk, trot, and lope on a loose rein, and relaxed
--stop softly and immediately from any speed
--back up freely when asked
--give to the bit in any direction, and break at the poll
--neck rein and direct rein
--be able to sidepass and two-track
--can execute a turn on the fore, or a turn on the haunches
--simple lead changes and (possibly) flying lead changes
--can perform “perfect circles” with little help from the rider
So for now, just work on basic riding. Get really comfortable being able to control all of your horse's body at any gait. If you can do that, and get comfortable with that, the barrel racing will come easier.