I would prioritize a good fitting saddle, a quality pad, a bridle set-up that works well for you and your horse. If you do the training slow and correct, there is no reason you can't run her/him in a plain snaffle. I personally prefer to run in my hackamore for casual shows, and when I was really competitive, I would run in my Myler Combination bit. That's just what worked for my horse. If I were to do it all over again, I would've tried a lot harder to get him comfortable running in a plain snaffle, rather than relying on leverage.
As for leg protection, I would prioritize getting protection for your horse. As soon as you start working on turns with speed, the opportunity to injury increases. I prefer my SMB Elite's and my SMB bell boots. Even if your horse isn't shoed, you may still need to invest in a set of bell boots. My horse is not shoed, yet he overreaches during runs and tears up his heels if he isn't wearing his bell boots. I personally just use boots on the fronts, but some people use boots on all four.
I don't really run with a breast-collar, but do have one available in my car whenever I do run in-case of saddle slippage.
Helmets are always a really good idea when dealing with horses, and they should be more prevalent in barrel racing and the like. I have seen too many horses slip around a barrel and throw their riders hard onto the ground, and many of them were not wearing helmets. Also, I would invest in a good sports bra to compete in, as I've had problems with my regular bras getting caught on the horn. I don't know how it happens, but it has happened, and it can really throw off your run.
Toofine - 1998 Half Arabian
Minnie - 2013 Morgan