The best way to thank my research supervisor?
You have the options already mentioned: sending an email or visiting them in person to say thank you. 

However (like me), you might be a little unsure of what to say when speaking face to face. In this case, a handwritten note or card would be a nice touch, as it's more personal and permanent than an email. 

You might want to ask for a reference at a later date, nearer to when you submit your applications. As Prof. Santa Claus says, if the project went as well as you describe, I see no reason why they would refuse.