Hello, I am somewhat new to programming and I am trying to write the code for an arduino uno. I would like the two dc motor to just run continuously the whole time. Then I would like the button when pushed to turn the servo from a 90 degree angle to a 0. Finally if possible could this program run without being connected to my computer, and if so how would I connect a battery to the circuit to achieve this.
This article may help you to get started with the DC motors and connecting a battery supply. Remember that the motors and servo will draw a fair amount of power, so the batteries may not last as long as you like.
For the servo and button, here is another article that may help.
Next step is to make a wiring diagram showing the devices you want, plus control (Arduino, [edit] DC motor driver {/edit]) and power (power supply), which will indicate how your program will control your devices.
This can be done with a MOSFET(s). Here is a simple circuit that will do it for you, You can parallel the motors and use only one MOSFET,
Depending on the MOSFET D1 may not be needed. The MOSFET would have a avalanche rating. You want the Vgs to be 3V or less on the MOSFET.