I am brand new to Arduino...
Start here, especially the General Design section. I would also suggest trying just one piece at a time, and start with the simplest examples for that piece. Gradually work your way through enough examples to the point where you feel comfortable with the capabilities of the piece and the interfacing software. Ask questions about specific examples or features, and you'll get better and quicker responses.
What sort of power supply do you think would be required for the main box, 4 servos, box with an on/off switch but sitting idle for perhaps 6-8 weeks?
You must look at the current requirements (milliamps or mA) for each piece. There are usually at least two values: one when it is fully active, and one when it is inactive or in standby. Add up all the active numbers and multiply by at least 1.5 to get the max current required. Also add up all the inactive numbers to get the standby current.
You will need a either a mains power supply rated for your max current, or a battery that can supply the max current at your minimum required voltage.
When using a battery, you will also need to use a regulator or DC-to-DC converter. There is a regulator on the Arduino board that can take 7V to 12V to provide 5V for the rest of the board. Your battery would have to provide the max current and stay above 7V. This usually eliminates 9V batteries.
Or you can get a converter module (which has a voltage input range) that can handle the max current. Some converters can boost a lower battery voltage up to 5V and/or drop a higher battery voltage to 5V. They also have an efficiency rating, which determines how much power has to be drawn from the battery to achieve the power used by your pieces. (Power = Volts * current).
To determine the required battery capacity, you have to convert the standby time (6-8 weeks) into hours. Then multiply your min current number to get the amp-hours required from the battery. For example, if it has to run for 10 hours, and it takes 10mA in standby, then 10 * 0.010 = 0.100Ah, or 100mAh. That would be the required battery capacity.
That's when you look more closely at how to power-down the pieces. Some, like the CPU, have different standby modes. Other times, you may choose to switch a piece off with a FET or an Arduino pin (watch those current ratings!).
As always, ask specific questions!
Cheers,
/dev