I had a similar problem connecting my Galaxy S3. The drivers were not installing properly. There were messages saying the USB device as malfunctioning. Despite being able to charge the phone, it was not being recognised as a mass storage device nor offering me an option on the phone as to what I would like it to connect to the PC as, until I changed my USB cable.
I had read some posts that had suggested this. After trying all the phone and device manager settings from debugging to reinstalling and updating drivers, this is what worked: trying a new USB cable. The drivers installed themselves. Then, I was able to select Use as Media Device from the phone notificaton menu. I double clicked My Computer, and opened Phone to display media. I was also able to select USB tethering from phone settings to allow connection to the internet, which is what kicked this all off in the first place.
Hope this helps some others.
What I also discovered during this process was how to set the phone up as a wi-fi hotspot. I just followed a neat youtube video detailing this, and was able to connect my laptop to the internet while I sorted out the rest of my problem.