This week I have noted we have available a new start page for Office 365 that it’s designed to be touch friendly as you can see (Note: Of course, each user can change the start page and choose to use OWA inbox, OWA calendar, etc): https://portal.office.com/home
If you want to change the Office 365 default start page and use another one, just follow these steps:
Click the settings icon and then the Office 365 settings option in the settings menu.
In the Office 365 settings page, click the “Start page” link and simply select the start page you want to use. Save your changes and you are done.