So I am trying to use this circuit to read in voltages from the wall and input it into the Arduino Uno. Would an F-1X transformer work for this application?
A transformer is required for safety (to isolate from the high voltage input). A common doorbell transformer can be directly connected to AC; if yours is NOT a doorbell transformer, it may lack internal protection (fuse link, resistance) against short circuits, and an external fuse and maybe a limit resistor are recommended.
The Arduino analog inputs have a permissible range of 0 to +3.3V when using the builtin 3.3V regulator as analog reference, so R2 and R3 sets DC level, and R1 and the S.A.T. (select according to test) resistor will attenuate the input voltage. I'd aim for 2V peak-to-peak, indicated by AC voltage on the S.A.T. resistor of 0.7 VAC. Don't trust the transformer ratio (because doorbell transformers, while functional, are not always accurate).
There's no need for diodes, the AC is easier to read without adding complications.
It highly depends on the accuracy of your reading and as you are using a 36:1 transformer means that you will have a greater uncertainty compared to lets say a 20:1 transformer.you are also using a 100 mega ohm resistor which in best case scenario has 5% uncertainty which means it can be anywhere from 120 to 80 mega ohms which is a huge difference.I suggest you measure the resistance of your resistor accurately rather than focusing on the transformer as the transformer has a much smaller impact on the accuracy of your reading.Based on the manufacturing website "Input Voltage (recommended) 7-12V" so if you work only with a 120V AC input then the best transformer would be (120/7 to 120/12) which I recommend the first one as it is on the lower boundary of your input tolerance and connecting the output voltage of the rectifier directly to your input,but if you really want to use a resistor to be extra sure about your board remaining safe then you can use a 1:1 potential divider with as low as a value as would feel safe with (using two 1K ohm resistors should be more than enough) and then just multiplying the obtained value by 2.I can help you more if you provide more detail on what is the range of the voltage that you want to measure and how accurate you want it to be.