Alright, first things first. You say, "...for a travel site that is for Desktop, iPhone, and iPad too...", to which I would say, you cannot (should not) have the same site for desktop, phone, and tablets. When it comes to choosing a date, this becomes all the more complicated because your desktop calendar may just fill up 80% of the phone screen, so you'd never want it to be there all the time.
Even for a desktop, the idea of having the date picker accessible by clicking on the calendar icon or focusing in the input field is to show the huge calendar only when required, thereby saving precious real estate. The tablets have a smaller screen size and hence the same principle applies.
So, showing the calendar by default in order to save one mouse click is definitely not a good idea.
If you desperately want to save clicks and real estate, try using a slider widget, or dropdowns, as some sites use. The slider/dropdown can be always visible consuming less area than the calendar. These could also be used across all the three kind of devices we are talking about, so helps both your objectives.