The RGB LED is wired to pins 9 (blue), 10 (green) and 11 (red), with a 330 ohm resistor between each pin and the LED. The colors should be as following, and need to cycle in this order. White, red, blue, green, aqua(green/blue), yellow(red/green) and purple(red/blue).
Well, there is no analogWrite() method that takes 3 pin numbers and a color name, so you have some work to do.
You need to connect your LED. with resistors, to the three pins, and create a sketch that contains nested for loops that cycle from zero to 255. Write the three values to the serial port. When you see a color you like, make note of the numbers.
Then, write a sketch that detects when a switch has been pressed. When the switch changes state, to pressed, increment a value. Do the same for the other switch, but make this one decrement a value.
Then, create a switch statement, in a for loop. Each case should call analogWrite() 3 times, once for each pin. Each case should set the pins to one of the colors you liked.
Finally, combine parts of the sketches to create a final sketch. That sketch detects switch presses and increments or decrements a value. It contains the switch statement that has cases that set the colors. The switch variable is the value that the switches cause to be incremented/decremented.
Since this is homework, presumably you are supposed to learn something in class that you apply in the homework. So, you need to try something. If you have problems, we'll help. We don't do homework for you, though.
Encouragement, hints, suggestions, links, some answers, if you ask the right questions, etc. are available, but not a completed program. You wouldn't learn anything from that, and it might not even work with your hardware.