Drupal - How to track and display selected user activities in a site?

There are a few modules that could help you out (quotes included are from their project pages):

• Activity:

  ... keeps track of the things people do on your site and provides mini-feeds of these activities in blocks, in a specialized table, and via RSS. The module is extensible so that any other module can integrate with it. The messages that are produced are customizable via the admin interface and are context sensitive.

• Heartbeat

  ... displays user activity on a website. This module is an API to log activity. The logged data contains message structures, attributes and variables. Once the activity messages exist in the database, they will be parsed and build into activity streams.

• User Activity

  ... calculates an activity index based on a user's logins, page impressions, created nodes and comments. The weight for each value is adjustable by an administrator. It provides an activity-o-meter on the user's profile page, a block with the most active users and an overview page of users sorted by activity.

• User Stats

  ... provides commonly requested user statistics for themers, IP address tracking and Views integration.

Which one(s) you choose really depend on what exactly you want to accomplish (use Views or not, display only post info or login stats, etc). I'm sure there are more out there that I am missing.

Activity stream modules (on Groups.Drupal.org) contains a comparison about the functionality and architecture of some activity modules.

For simple use cases, core provides the often overlooked Tracker module, which is what Drupal.org uses (although, if you're on Drupal 6, the Tracker2 contrib module is more performant, and made its way into Drupal 7).