Login to the MOP as a global admin. Click on the Admin drop down on the far right by your name and click "SharePoint"

On the next page, you will see a list of all your site collections. Look at the last column and you will see what version of SharePoint that Site Collection is on. If its on 2010 and you want to upgrade it, just put a check in the box right before the URL, and on the top ribbon click "Upgrade"

When you click Upgrade, a small window will pop out from the bottom of that button and you want to select "Site Collection Upgrade Settings"

You will then have a window pop up. You want to click the link that says "Link to the upgrade page"

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.