There are many
different ways to do what you describe, depending on what exactly you plan to do with the saved sessions, how many you think will be captured, and so on.
Without writing any code at all, you could do this with a combination of Fiddler's Filters
tab and the Tools > AutoSave
Using FiddlerScript, you could save in other formats to the file of your choosing; FiddlerScript is simply JScript.NET, so you can use any of the methods in the .NET Framework (e.g. the System.IO classes) to save data. Or you can use the appropriate methods on the Session objects.
Check out the Telerik Platform - the only platform that combines a rich set of UI tools with powerful cloud services to develop web, hybrid and native mobile apps.