I'm wondering if there is any analytics tool more sophisticated than google analytics which offers features such as user clicking and navigating behaviors. By which I mean, with such tools, I can see how users actually clicked and navigated across my site.
I'm not sure if you have a programmer, but assuming you do this advice should work.
For a low traffic site, just log everything to your database.
As you grow, logging directly to a database won't scale nicely. I've had good luck using a message queue (eg. RabbitMQ) to log clicks and a cron job to process them in a batch. Inserting a batch of several log messages into your database as 1 transaction is much more efficient than making every page load insert a row into your db.
I don't know any free services for that. What you want are called heatmaps.
One paid is http://www.crazyegg.com/
But as suggested by @Gattster you cans develop one on your own. Just send each 0.1 seconds the x,y coordinates.
Well Click Heat is free and shows you exactly where visitors are clicking on your website, it can tell you how to better optimize your site. For example if a graphic is not a link yet you see a lot of people clicking it assuming it is you can turn it into a link
This is pretty basic but again it's free and shows more click information than Google which only shows clicks on links.
For paid click tracking crazyegg.com is good or clicktale