For a self-balancing robot (two wheels?) with possibly a Bluetooth addition, the easiest/best option is probably the Uno, or perhaps the Leonardo (which is like an Uno with better USB support.)
The Due adds more processing power, which you will not need, and it comes with the significant limitation that it only works with 3.3V I/O, not 5V. Also, it's more expensive.
The Mega adds more peripherals (pins for connecting things) but the Uno/Leonardo already have enough pins for driving the sensors and actuators you need.
I'd recommend the Leonardo, plus a motor shield, plus a gyro/IMU that talks I2C, plus perhaps a Bluetooth shield, and that would get you all the way there.