I have a few thoughts on this but I can see problems with both. I don't need 100% accurate data. An 80% solution that allows me to make generalizations about the most popular domains I'm routing users to is fine.
Option 1 - Use PHP. Route links through a file track.php that makes sure the referring page is from my domain before tracking the click. This page then routes the user to the final intended URL. Obviously bots could spoof this. Do many? I could also check the user agent. Again, I KNOW many bots spoof this.
Option 2 - Use JavaScript. Execute a JavaScript on click function that writes the click to the database and then directs the user to the final URL.
Both of these methods feel like they may cause problems with crawlers following my outgoing links. What is the most effective method for tracking these outgoing clicks?
The most effective method for tracking outgoing links (it's used by Facebook, Twitter, and almost every search engine) is a "track.php" type file.
Detecting bots can be considered a separate problem, and the methods are covered fairly well by these questions: http://duckduckgo.com/?q=how+to+detect+http+bots+site%3Astackoverflow.com But doing a simple string search for "bot" in the User-Agent will probably get you close to your 80%* (and watching for hits to /robots.txt will, depending on the type of bot you're dealing with, get you 95%*).
*: a semi-educated guess, based on zero concrete data