I wrote something similar in Greek but didn't write how to move from Google to ownCloud. The 3 things you should do are, files, contacts and calendar.
The ownCloud application can be downloaded from Google Play and costs 0.79E. You can also download it from F-Droid market.
This program is amazing. It works just like the others. You can even upload all your pictures from your Android phone directly to your ownCloud instance.
Alternative application is FolderSync Lite. It seems to work fine with WebDAV with ownCloud.
In ownCloud 8.0.x, calendar isn't enabled. So you have to enable the Calendar app. Log in as "admin", click on "Apps+" and activate the Calendar app.
Log into your Google Calendar and go to the settings hidden behind the cog wheel icon in the upper right corner. Click on "Calendar" at the top. Look for a "export calendar" link. You will get a ZIP archive downloaded that contains an ICS file. Unpack that ZIP archive. Navigate to the files section on your ownCloud in the web browser and upload the ICS file to any folder using drag-and-drop. Then click on the file in the browser and confirm to have that ICS file imported to your ownCloud calendar. Now your calendar has been copied to ownCloud.
Finally remove the appointments from your Google calendar and stop the sync from your smartphone. Neither can you remove the Google calendar from Google nor from your phone. Make your ownCloud calendar the new default on your phone.
Install CalDAV-Sync on your Android phone (it costs 2.59E). Run the application and set up a CalDAV account. The CalDAV URL can be found in the ownCloud web interface when you navigate to your calendar and click on the cog wheel icon in the lower left corner. It looks something like https://demo.owncloud.org/remote.php/caldav/. You can also get the link from a specific calendar (just check the link next to the calendar you want).
A replacement for a free program is aCalDAV from F-Droid.
In ownCloud 8.0.x, contacts isn't enabled.In ownCloud enable the contacts app. Log in as "admin", click on "Apps+" and activate the Contacts app.
Log into your GMail account contacts and click on "More" and "Export". Choose to export all contacts in vCard format. Navigate to the files section on your ownCloud in the web browser and upload the VCF file to any folder using drag-and-drop. Then click on the file in the browser and confirm to have that contacts imported to your OwnCloud contacts.
And finally remove your contacts from GMail and stop the sync.
Install the CardDAV-Sync app on your Android device (it costs 1.90E). Run the application and set up a CardDAV account. The CardDAV URL can be found in the ownCloud web interface when you navigate to the contacts app and click on the cog wheel icon in the lower left corner and then on the globe icon. It looks something like https://demo.owncloud.org/remote.php/carddav/addressbooks/test/contacts.
A good replacement is Contacts+.
There is another pretty good solution for syncing an Android phone with a CalDAV/CardDAV if you don't want to spend money. It's called DAVdroid and it is open source, so you can download it from F-Droid.