I have a Droid 4, but this problem has persisted through a couple of phones ( a Droid 2 and 2 Droid 4s). I have tried multiple calendar apps, thinking the problem was the app. (Touch Calendar and Calengoo)
My contacts and calendar do not sync to google. When I got a new phone and download contacts and calendar, I have to go phone to phone. Then, all contacts are duplicated- old information from gmail is also downloaded. When I update contact information on my phone, my gmail contacts do not change.
If I create an event in google, it is downloaded to the phone, but events on the phone are not uploaded.
I have tried multiple hard resets, multiple phones, in fact. The problem has persisted for a few years, I am just finally annoyed enough to ask for help. 
I've been to the verizon store locally when I got my new phone to get the contacts transferred, but they couldn't figure out why my google account wasn't listening to my phone.
I am not sure if it is connected, but every time I connect my phone to my computer to move files, the back up assistant site opens in the browser and starts playing the commercial... also very annoying... is there any way to stop that?
I am not very tech-savy, so baby steps are needed, please!
thanks!