What you need here is Ajax. You will need to read about how to use WordPress Ajax (link).
When your visitor click on select field to choose sorting, you need a JS function to handle the data about how to sort and where the visitor in, then a js function to make ajax request to server.
At server here (I mean your theme function) you will need a function to handle request and send the response to client. The handler function here is a WP_Query loop to loop your post according to the order that your visitor chose.
Sorry, I don’t have enough time to provide you the code, but I think for a scripter/coder/developer, the direction is enough.