First I would recommend that you check out the "sticky" on this barrel forum here: Barrel Racing Exercises and Drills.
We all contributed to it so that it can be a place for newbies to read through.
But to answer a few of your questions up front...
It can take anywhere form 6 months to 2 years to train a barrel racing horse from start to finish. So the key is to be patient. And go at the horse's pace.
If you try to go faster than the horse is ready for, you will create bad habits that are very hard to fix later on, and you could potentially make your horse soured on them. When horses are pushed faster than they are ready for, they get confused because they don't know what is expected of them. That confusion is what leads to the horse getting soured and frustrated with the rider.
Flying lead changes are not necessary to train for barrels, but it is helpful. At the minimum, your horse needs to know simple lead changes.
When your horse is a finished barrel racer, they will switch leads on their own, because they know what to expect. But when you are training them, it is often best to cue them so that you can set them up for a successful turn. (They can't turn a barrel properly if they are on the wrong lead.)
If this is your first time barrel racing, I ALWAYS suggest that you work with a trainer who can show you hands on. You can read all the books you want, and watch all the videos you want, but hands-on training cannot be replaced.
You wouldn't want an airplane pilot to fly the plane because he feels confident because he read all the books and watched all the videos .... but had zero hands on training .... would you? So you should never expect to learn something you've never done before without having hands-on and in-person training.
But first read through that sticky thread I posted. We've got a lot of good info in there. Feel free to ask any more questions though!