Barrel racing is a blast, but it isn't a "cookie cutter" for everyone. I started barrel racing when I was 4-years-old so I was fortunate to grow up with it. I have a cousin who's been taking English riding lessons and she took lessons for 2 years before ever even competing in her first show (jumping, at a trot).
Everyone is going to learn at different paces. So it's really hard to give you a time frame because it is going to vary so much.
First I would encourage you to check out this thread: Barrel Racing Exercises and Drills.
It's got a lot of great information for beginner barrel racers.
How experienced of a rider are you? You said that you want to start riding "again" so this would mean you already have some riding experience. Either way, I highly suggest you take lessons from a trainer because it is very important to control and cue your horse effectively.
What horse are you going to be using? It would be wise for you to purchase an older (over 10 years old) horse who is already trained for barrels and suitable for someone just starting out. Yes, it is possible to train your horse for barrels as you are learning, but it is not the ideal route. Keep in mind it takes 6 months to 2 years to correctly train a horse to barrel race; usually closer to that 2 year mark.
As far as what you need
, well just you and a horse.
But you might also want
a barrel racing saddle, breastcollar, bridle, and protective sport boots. Obviously a pickup and horse trailer if you plan to haul to the trainer and/or barrel races.
The key to learning to barrel race is to go SLOW and CORRECT. If you try to go faster than you or your horse is ready for, you are going to create horrible bad habits that are difficult to fix. And the good riders out there will spot you a mile away and know that you have no clue as to what you are doing. (Hence why working with a trainer is a must.)
But barrel racing is a lot of fun, and you meet a lot of great people while doing it too.