Find a core system that interests you and learn about it

Needs revision -- Last peer review: 13 Mar 2016
Steps: 

Here's a list of Drupal 8 core initiatives and Drupal core systems with links to documentation and additional information on drupal.org and groups.drupal.org. DrupalLadder.org has a growing library of lessons for learning about these (see below). As you work your way through the documentation, please consider sharing your knowledge by contributing a lesson to DrupalLadder.org:

D8 core initiatives
Overview of Drupal 8 core Initiatives
Drupal 8 Initiatives- announcements and discussions on G.D.O. (Groups.Drupal.org)

Lessons
DrupalLadder.org also has a growing library of lessons for learning about Drupal core. Check out these lessons by clicking the Lessons tab, and selecting "Core" Project type in the filter:
Lessons | Learn Drupal

Drupal 7 Core Systems on API.Drupal.org
ajax system (ajax examples)
base system
cache system (cache examples)
cron system (cron examples, queue examples)
database system (database examples)
database update system / update.php
entity system
field system (additional Field APIs)
file system (file examples)
forms system (form examples)
image system (image examples)
install system
language system
mail system (email examples)
menu system (menu examples)
node system (node examples)
theme system (render examples)
token.inc (token examples)
xmlrpc (xmlrpc examples)/

Issue Queue Access
Drupal 8 Issue Queue

Other Places where you can search issues. These link back to the Issue Queue
Drupal Core Mentoring