Dealing with my own young horse ATM (she had basic training a while before I got her, but she was left in the pasture for a long time, and decided she'd rather not work), and here is what I've learned from others. I'm training her with the bit, like you, in case I need to sell her; I guide her with my legs and neck-reining, though, so it's both easier for her and I have more control (legs+reining means more and gives more command options than just one alone). Most of this is from the Buck Brannaman 7 Clinics DVDs (not affiliated with me--just a happy customer); do check them out--you'll learn a lot on how to train a horse to extreme performance levels with pure positive reinforcement.
First, get her to drop her head and allow you to pet her all over her face. Start on the forehead, down her nose, her cheeks, etc. She may well be familiar with this from other training already--pet her regularly, and as a reward for doing good. Next, she needs to allow you to handle her nose and ears. Run your hand quickly, not abruptly, over her muzzle and under her chin. She might throw her head up--you do it quickly so you are in and out before she has time to really fuss; keep working at it, and she will learn it's OK and allow you to handle her. Do the same for the ears; never bat the ears, just run your hand over them quick and light. Eventually, you should be able to handle her muzzle and waggle her ears without complaint.
The next step is to work on opening her lips. You should be able to slid a finger into the corner of her mouth and lift her lips. Don't put your finger in too deeply so you aren't bit. She should remain completely calm. Next, take your lead rope and slip that in where the bit will go. It's softer than the bit, so it's easier for her. Once she's completely calm about all this, you can finally put the bridle on.
When you put the bridle on, do not leave the halter on. Untie the halter, then slip it through her mouth under, then over, the bit--just like you did with the rope before. You can do this in reverse when you are done riding. Eventually, you'll probably be able to trust her to stand for you while you switch between them, but even then you run the risk of a stray dog scaring her, or something, and having a runaway horse. When you take the bridle off, don't drag it over her ears--hold her ears and bend them so they slide under without even being bumped.
Adjust your headstall so there are no wrinkles in the corner of her mouth; a lot of people want that slight pressure there and think it makes them more responsive--it just makes your horse more numb to pressure, and annoys them by pulling on their cheek. When you apply even a light amount of pressure, it's more painful on their cheek as well; it should just rest in the corner of her mouth, and you should be able to turn her by just lifting a rein--no pulling. If she starts getting the bit under her tongue, then you can pull it up for a couple days so she can't do that until she accepts the bit on her tongue--it can really cause problems if this becomes a habit. Once she stops trying to do that, lower it again; if she can't figure it out even with the bit higher up, you'll need to use a bit of string over the bridge of her nose and tied to the bit on each side to pull the bit up to the top of her mouth.
As far as bits go, I use a full-cheek copper snaffle. No rollers, sharp spots; copper to promote a moist mouth. I use a western bridle; no leather over the bridge of her nose, and no chain underneath her muzzle. Don't tie her mouth shut--you can tell when they are worried or in trouble because they work their mouth. It's far better to solve the problem than ignore it and stop the symptoms--that's a great way to get into a rodeo.
Good luck training her!