Display twitter tweets for own account in drupal 6
The easiest way is definitely to install the Twitter module that you have already discovered. I used to do this on my website as a way of pulling in tweets to my blog. You will only need the read-only configuration for the module, and you may find it easiest to use the Views integration.

If you'd rather code your own, the QueryPath module allows you to write jQuery-like requests directly against Twitter's API, but you'll have to write your own module. (You can start with the sample modules that come with QueryPath).