What I want to happen: Pressing button 1 will initiate the performance of one series of servo operation, pressing button 2 initiates another, and so on. There are 7 buttons in all and 4 servos. One servo is a Vex continuous rotation 269, the other three are hitec 422 servos. I should be able to program it without a hitch, but I am not good with electrical stuff. Here is the diagram:
The batteries to the arduino will be a battery pack that fits the arduino dc jack. Also, I really dont know how to determine how many batteries to use. Also, does the arduino 5v need to be wired in there somewhere?
1, I would indeed include the Arduino 5V in there, as I'd use that as the +ve side of your switches, not the +ve side of the servo power as you have shown. Reason I say that is that servos can run at 6V, and if you used 4x1.5V batteries for the motors then you would be taking your Arduino data pins to 6V.
2, Conventional forum wisdom is to budget 1A of current per servo, or at least per servo that operate together. At worst that could mean providing each servo with a pack, at best one pack as you have shown. It's the current that's the issue, and if it's too little you'll likely see the servos vibrating rather than moving nicely. Of course, you can easily measure a servo's current with an ammeter in series with the power supply.
3, You can make your life muuuuuuch easier by not using external resistors on those switches. Look at this example which uses the internal pullups. Makes your wiring way simpler. Only thing to be aware of is the logic: looks like your resistors are pull downs, so i/o pin is normally low and pressed is high. With pullups (there are no internal pulldowns) the logic is reversed: pin is normally high and pressed is low. It's easy to take care of that in code, so no biggy, and it is in any case a better electrical practice.