Hi Adrian
I think your information source is talking about code efficiency and not about code reduction. Code efficiency means writing code which is readable, portable, consistent and scalable. Using mysql_result() is readable, but that's about it. It returns data from one single cell. This can be useful in some situations, but is not really good for you, I suspect. I recommend mysql_fetch_array() as an all-purpose, flexible data call to MySQL.
The 'NEW WAY' you are talking about, isn't really new. It just indicates an object-oriented approach. You need a DB data-retrieval class and you then create an object from the class. This is a lot easier than it sounds, but it does take a little getting used to.
I suggest you play around with the various PHP-MySQL functions and then maybe think about applying some OOP principles once you're used to what PHP offers in terms of MySQL.
HTH
Norm