Since you are using set data, you really don't even need to use a microcontroller at all. But, using the arduino, check out this library: Arduino Playground - SevenSegmentLibrary, you will have to modify that because it will not have the patterns for the degree symbol or F and C.
An LED display is just 7 LEDs with a common anode or cathode (Seven-segment display - Wikipedia). The display you cite is a common cathode type.
You need 7 330 ohm resistors. These will go between the a-g segments and the arduino. On the arduino, you wire all a's to one pin, all b's to one pin, and so on. Then for each display, you put the common cathode to an individual pin.
You set your a-g outputs as you want them for one display and then drive the common cathode for that display low to turn on the LEDs.
These are the patterns you need:
7 = A,B,C
8 = A,B,C,D,E,F,G
2 = A,B,G,E,D
6 = A,F,G,E,C,D
F = A,F,G,E
C = A,F,E,D
DEG = A,B,F,G
You have to cycle through all displays repeatedly. You do it fast enough that the eye doesn't see the flicker. Set a toggle in code for your button to keep track of the two states (F and C) and then just light up the displays.
You would need 12 pins total (7 for the segments, 4 for the digits, and 1 for your switch.) If you need to save on pins, you could just hardwire the displays with your two different states, and turn them on using only 1 pin for each state (so 3 pins total.)