How do I easily switch between windows rather than applications in GNOME Shell?

You can use Alt+` (the key above Tab) to cycle between windows of the same application.

You can even mix Alt+Tab to cycle between application and Alt+` to cycle between windows of the selected application.

There is an exposition of the rationale in this blog post from Canonical's Didier Roche. It's part of a series discussing the development of gnome integration in Ubuntu 17.10.

Alternatetab is just the Extension you were looking for!

Follow the Managing Extensions instructions here in order to activate it (you should read the text before that section too though). As it is a pre-installed Extensions you will only need to activate it (unless you have removed it, in which case you will need to follow the Installing a new Extension section).

So now upon pressing ALT + TAB and then pressing TAB without letting go of ALT (until you have selected the window you want to switch to - then simply release both keys and it will put that window on top) you should be able to switch between individual windows as opposed to applications!

As someone still quite new to Ubuntu, I found the other answers confusing. After a bit of digging I found a simpler solution, as I also want Alt + Tab to toggle through all the open windows, not just through the open applications.

In Ubuntu 18.04, you just need to go the Keyboard settings as shown here

Then set Switch windows to Alt + Tab. It will automatically disable Switch applications which is assigned that key combination by default.

No extra software required.