Watch the Experts, then Watch Others
You can find plenty of videos to teach you all of the cool stuff that DevTools can do. Watch a few of them. I highly suggest starting Paul Irish's Chrome Summit Talk. Then just search search for DevTools on YouTube. Pick a few that look interesting. Watching these videos is the quickest path to get you moving on all of the cool features DevTools offers.
Read the Docs
Google provides a tremendous research on the DevTools site. It can be overwhelming how much there is to read, so here are a few links to start with.
- Tips and Tricks
- Keyboard Shortcuts
- Editing Styles and the DOM
- Using the Console
- Authoring and Development Workflow
Experiment!
Nothing beats working with it. Once you read or watch about something new, give it a spin. Try to use DevTools as your primary front-end code editor for a day. Make yourself use the Keyboard Shortcuts. Write some Javascript in the console. Check out the extensions available for DevTools and see if any of them can improve your workflow even more.