How would I go about programming such device in terms of software and language? Would I use HTML and make a web interface? Need advice so I can get started on this idea that I have.

do you have some specifics? hardware factsheet?
can you run any os on it as you like?

Web would be fine for the interface but have a look at the printer and card reader. Possible the work as a standard printer and emulate keyboard. But I'd say you will need some backend logic to deal with the reader at least.