You need an omni mixer
. The mixer works with your ESCs and radio to get three planes of movement(Forward-Backward, Left-Right, Spin).
The gyro you're looking for is a peizo gyro. You'll want a heading hold gyro(Or a Tail-Lock gyro, same thing). Its purpose is to auto correct for unwanted control deviations. For two wheeled bots, this means a straight, true line. For four wheeled bots, its a smooth turn. I don't know enough about omni wheel driven bots, but I'd imagine its somewhere in between(Anybody with more experience with omni wheels please fill in). If your bot is invertable, you'll want a method of turning the gyro off remotely. Gyros when flipped upside down get grumpy and get stuck in a infinite feedback loop(ie; an uncontrolled spin that wont stop until the gyro is shut off). Higher end gyros have the switch as a feature, but if your gyro doesn't have it, you can just get a dedicated switch
One last thing, you might want to look into dedicated omni wheels
. Regular wheels would cause huge friction losses if used in the configuration you're talking about. Just keep in mind that they're a bit heavy, and they're fragile. Keep them internal and be prepared to replace them when they shatter.