Michelle:
I am not sure what you are referring to by Old and New City. There is an area known as the New Town (which is over 150 years old!) and there is the Royal Mile area, each is within walking distance of the other, but the extreme ends of each may be a bit far to walk. It depends what you want to see, but buses take you quickly and easily wherever you are, even beyond these areas.
As for exploring the Highlands in a day, again, it depends what you want to see. You could get a bus tour to take you for a day trip, or if you have a car (which you won't need in Edinburgh), you could plan a day trip. The Highlands certainly are worth seeing for the scenery and some of the old castles, etc., but a day trip will only scratch the surface.