How do I get the exact share-count for my website links? I want to trace all the shares/likes/social mentions.

The best way to track sharing across the board, usually, is to use a link that tracks shares (such as a service like bit.ly). The downside to this is that tests have shown people like to see the long form link and don't trust shortened links. Otherwise, Shared Count or BuzzSumo is usually a good tool.

Of course, you can always use Google Analytics to track traffic, as well as Alexa to see where/how people are linking to your site (on a broader scale).

Lastly, there is also the option to manage your social media via a service like HootSuite or Buffer. You can set parameters to track certain links, your original posts, etc. This is also limited, as you won't get an exact count, per say... But you will be able to see every single share/comment/like and track it one interface.