That's a good place to start. You'll want a regulator for your 10V supply and a transistor/resistor/capacitor for each channel (group of pumps) for the low pass filters.
But you can get yourself up and running with the Arduino, learn to flash the onboard LED, get serial communications working, etc.
The Arduino plugs into your PC via a USB cable. The firmware is flashed over this interface (the "Upload" button in the Arduino environment). Additionally, it acts as a serial output when the board is running (search "arduino serial console"), you can use this to print messages from the board to your PC, and to send serial commands to the Arduino.