spietreser on "Update from 3.3.1 to 3.7.1"

Hi all,

I want to update a client's site from Wordpress 3.3.1 to 3.7.1 (latest version at the time of this writing.

Before I start just updating through the built-in updater from the admin area, I'd like to know your 2 cents on:

What is the best way to do this?
What are the possible pitfalls I need to watch out for and how do I avoid them?

Thanks! I look forward to reading what you can tell me about this!