Sounds like a good beginner project. It should be pretty simple to create... and there's probably a few different ways you could do it.
The way I would do it (which might not be the best) would be to create a "gateway" script to handle all of the links. Basically you'd call the script by passing the url as a GET string, like this:
<a href="gateway.php?url=www.blahblah.com">
Then, the gateway script will take the url var and dump it into a db. You could do a number of things here to, depends on what you want to accomplish with it though. You could just insert a new row into a table for each time a url was clicked... or, you could check to see if the url already exists in the db, and if so increment a counter for it... or, there's a number of different things you could do here. You could even keep track of IP addresses, and HTTP referers, etc.
As far as being "transparent" to the user, they will see the url as gateway.php?url=www.someurl.com, but I'm not sure if that's a concern for you. Also, I can't think of any other way to do it... but there's always more than one way to skin a cat.