Save a Data Stream to notepad on another Network drive wirelessly

Projects General Guidance
1

I need a little project guidence

I have an automatic transfer project that I have been working on for quite a while. It works, and is on automatic pilot. I have been experimenting with collecting the data, that I get out of the Arduino Uno and veiwing it in the Serial Monitor. Then I discovered 'Cool Term', a terminal program that allows me to monitor the data just like the Serial Monitor. Cool Term also has a feature that allows the data to be saved to a notepad.txt file, that I can save to a network drive. This is a very cool thing. I have to drag my laptop out to the garage to collect this data. I would like to be able to do this with out the laptop.
Here in lies where I'm at. I have an ESP8266-01, but I don't know how to make it perform what I need:

  1. create a file like cool term does and
  2. save that file to another computer on my home network

This is where I need project guidance. When I search I see how the ESP8266-01 can:

  1. use a cell phone to trigger a relay to control whatever
  2. send data to a web page
  3. many other things

But I don't see anyone saving data from an UNO, through an ESP8266-01, to a notpate file on another network drive. It has been suggested to me to just save the data to a memory device, then retrieve the memory device later. Well that is not what I'm trying to do. Can anyone point me in the right direction on this. Maybe there is another piece of the puzzle, I'm missing.
Stephen