How can I root my Samsung Galaxy S3? Should I defer the system update? I pre-ordered a Verizon SGS3, and it shipped to me before it was even available in stores. Since then I have not updated any programs that were Verizon-related.

If at some point I want to root my phone, does it then make sense to not update these programs? I have heard that in the past, certain over-the-air updates have closed holes which are used to root. Is this the case so far with this phone?
Update: Verizon's Galaxy S3 is locked and cannot as of now be rooted or updated to a custom firmware. See this XDA thread.

As far as I know there are no current pending security holes that can be exploited by circulating rooting apps at the moment.

Samsung does however not lock its phones if you buy them directly from them, i.e. you are free to install aftermarket firmware or modify the existing firmware via Odin quite easily (without having to hack/exploit/etc. anything). Some carriers, including Verizon however do indeed lock it.

If you have an unlocked S3 or as soon there's an exploit for the locked booloader:
CF-Root by Chainfire supports rooting stock firmwares of the S3 models GT-I9300 and GT-I9300T so far in a very easy way. It does not depend on an root exploit but rather modifies the system partition in another way that cannot be closed by a firmware OTA update. Just see the above referred blog post and the linked XDA forum thread  in there for more info.