I suggest you first take a look at the “onchange” JavaScript event handler. This event is triggered any time an input changes. In this case, it would be when someone selects from a drop down. You can have this listener attached to multiple drop downs where if any of them change, the event is triggered.
In the event handler itself, you will then read the three drop down box values. With them, you can construct a value you will then place in the date input field. In this event you can also determine if the field is a date input that is supported by the browser or if it is not supported and change the value you place in the text input (for browsers like Safari).
I suggest you try the link above which will link to W3Schools where you can then use the “try it” button to see how this event works. Once you see how that works, look at the JavaScript querySelector() method for getting access to the drop down boxes along with the date input. Then it is just a matter of setting the date input’s .value property to set the date.
I know it sounds like a lot, but it isn’t too bad. Just break the problem down. Get the event listener to work for your three drop downs first, then move on to pulling out their values and setting the date input value. Lastly then look at how you might detect if the date input is a date input vs a text input. 
