How to upgrade SharePoint Site Collections and MySites on an Office 365 Enterprise service plan
That will take you to your SharePoint upgrade page. All you need to do is click that blue box that says "Upgrade this Site Collection", and follow the on screen instructions. It will update automatically and once its finished, you will be on SharePoint 2013. You can go back to the Site Collections and do the next one. You only one to do one at a time.
As for the MySites, It should upgrade the users MySites automatically to the new SharePoint 2013, but sometimes, existing users will see that when they click SkyDrive, that it still takes the user to the SharePoint 2010 web interface. If this does happen, as seen in the image above, the user will see that same bar across the top and all the user will need to do is click the Start Now and then they will see the same image as above. Once the user clicks Upgrade this site, it should not take more then 10 mins to upgrade.