Consider the following combination of products:
4x 48mm Omniwheel for LEGO NXT, Servo
4x HSR-1425CR Continuous Rotation Servo
2x Lynxmotion Aluminum Multi-Purpose Servo Bracket ASBI-04 (Pair)
1x Arduino Mega 2560 Microcontroller Rev3
1x DFRobot I / O Expansion Shield for Arduino MEGA v1.1
1x 9-12V Battery for the Arduino and 1x 6V battery for the servos (or use a voltage regulator).
You would need to custom-make a base for the robot.
The wheels mount directly to the continuous rotation servo motor (remove the horn) and are kept in place with a screw. The servo bracket helps you mount the servos to a base.
There is no specific screw size - just so long as it fits through the hole. The brackets come in pairs and each mounts one servo. You can easily make the robot symmetric (it would look like the DFRobotShop Omni Rover). Using the brackets is entirely up to you - just trying to help.
Those are Mecanum wheels - the robot is slower than an omniwheel driven robot and the motors cannot be changed. It’s 8.5in x 5.5in. There are not many omniwheels on the market and the ones which are compatible with faster and more accurate motors are from Andymark (larger and more expensive). If these interest you, we can explain which motors would be compatible.
4x 48mm Omniwheel for LEGO NXT, Servo
4x DFRobot 6V. 180 rpm gear motors
Those wheels can accept either the servo or the gear motor. The servo rotates at up to 52rpm (no load).
Be wary of clone boards - the quality is not good. You did not indicate your preferences:
There are single DC motor controllers, dual DC motor controllers and the rare four motor controller. We would suggest two dual motor controllers.
If you want to use continuous rotation servos instead, you would use a servo controller, and most have more pins than you will ever need.
Closer to 30cm diameter
Do you want DC motors or continuous rotation servos?
Do you want plastic roller wheels or rubber (for more grip)?
Do you want two dual motor controllers or four singles?
There are many many parts to choose from and providing you with a list would only be one of many possible examples.
4x 48mm Omniwheel for LEGO NXT, Servo
4x HSR-1425CR Continuous Rotation
Use the external power input on the shield to power the servos (that’s what it looks like anyway).
You’ll need to design your own base to mount the servos, Arduino, battery etc.
The shield which comes with the Arduino seems to have a screw terminal which can likely be used to power the V and GND lines.
For servos you won’t need a motor controller card. You would need 6V batteries (4x AA rechargeable or alkaline, or a battery pack).
No - you would only need one 6V supply for all servos. A popular IR distance sensor is the Sharp GP2 series and the corresponding cable. There is no specific compass, so choose one based on an interface method you are comfortable with and is within your budget.
Sorry, didn’t catch that. You need an IR receiver which operates at the same frequency as the ball.
There you have it 
You will need a motor controller such as this one in order to drive DC motors: RB-Dfr-04
Continuous rotation servos have their own integrated controllers.
Also, you could simply get this complete Arduino kit: RB-Rbo-125
