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.
