This project can be useful for you if you have a website that you can use to keep the data. The common way is to set the ESP8266 to server mode and connect to it from the internet by entering your router static IP address in a browser. My internet provider don't have this option. In this project the ESP is set to station mode and it sends several bytes of data to my website. I can browse at the data from the internet by logging to a page in my website. The idea is taken from here: https://thingspeak.com/ The idea is to request a file and to add the data using URL parameters. The PHP file records the data on another file which I named a2.php. I added a suggestion for a PHP script http://moty22.co.uk/wifi.php . To see the data log to page http://moty22.co.uk/a2.php .
In the code enter your SSID and PASSWORD of your router. Also change the name of my website to yours. You can use my website to test your project, http://moty22.co.uk/a2.php but remember that if more than one person is using my website at the same time they are likely to read each others data.
The internal connections of the Arduino show the serial data coming out of the Atmega only. Monitoring the data both ways will give better view. One way to achieve that is by connecting the Tx output of the ESP to the Rx input of a serial to USB converter module and setting the Serial Monitor of the Arduino IDE to read the port of that USB.
A disadvantage of this system is that you can't access the ESP directly from a browser on the internet. This code is set to transfer the data to the website once every 60 seconds.
Changes to the code can allow for transfer of more data and also data transfer both ways.