Date exposed filter default value
This can be done through contextual filters in views. 

Navigate to edit the structure of your view, click advanced and add a contextual filter. Here you will select the date field and "Apply"

On the next page, under "WHEN THE FILTER VALUE IS NOT IN THE URL" you want to "Provide default value" of type "PHP Code". The code you want to use is fairly simple, but depends on your date/time setup. you should have something like this in the php code field:

return date("Y/m/d"); 


Good Luck!