If there is a standard, it would be that 0 to 440 pixels wide would be for mobile, 440 to 760 would be for tablets, 760 to 1200 is for computers and above that for wide screens. However it's not really a standard, more a general guideline as a specific design might "break" at different widths, for example 790px. Those are the widths used by most responsive WordPress themes.
You need to consider the way the site is used; mobile is generally used more vertically and tablets horizontally, but both will use a touch screen, which you need to consider when sizing and spacing your elements. Of course, both orientations have to work regardless of the size.
As for designing separate sizes individually, it hasn't been done this way in a few years. Do you remember at the beginning of smart phones when a mobile site was m.example.com? That required separate sites that both needed to be updated individually at every change or post. Nowadays, the recommended method is to use css media queries to make sure that the design is optimized and will adapt no matter what width the site is viewed at. For example, you could make elements lined up horizontally change to be vertical on smaller screens, change the font size, remove sliders or other bandwidth demanding elements, make the buttons bigger, and so on, and make to changes below a certain width.