I would recommend using jQuery UI's datepicker tool. It looks great and has support for a lot of customization options (min and max dates, date formatting, etc.). You can see a demo and find the full documentation here: https://jqueryui.com/datepicker/
As for storing the data, I would recommend using a free format question. There's no one right answer to what variables you ought to use, though - it just boils down to what you need done with the data. I explored three options here: https://www.sawtoothsoftware.com/forum/11744/date-picker-in-free-format-question
I can help write the datepicker code for you if you can offer specifics about what functionalities you need it to have and how you want the data saved.