How do you 'redo' changes after 'undo' with Emacs?

Emacs 28 adds a redo command (called undo-redo).

If you want to have more typical undo/redo, the following commands can be used.

(global-set-key (kbd "C-z") 'undo-only)
(global-set-key (kbd "C-S-z") 'undo-redo)

To access this functionality (with the ability to use non-linear history too, see undo-fu which is a thin wrapper on emacs built in undo).

By default, redo in Emacs requires pressing C-g, then undo.

However it's possible use Emacs built-in undo to implement a redo command too.

The package undo-fu, uses Emacs built-in undo functionality to expose both undo and redo.

Edit, undo-fu has now been integrated into evil-mode (if you're an evil-mode user).

Example:

(use-package undo-fu)
(use-package evil
  :init
  (setq evil-undo-system 'undo-fu))

Doom Emacs users, I hope you've scrolled this far or searched for 'doom' on the page...

• Doom Emacs breaks the vanilla Emacs redo shortcut: C-g C-/ C-/ C-/ etc (or C-g C-_ C-_ C-_ etc) ...and instead that just keeps undoing.
• Doom Emacs also breaks the undo-tree redo shortcut mentioned in one of the other answers as being useful for spacemacs etc: S-C-/ (AKA C-?) ...and instead that throws the error "C-? is not defined".

What you need is:

• to be in evil-mode (C-z to toggle in and out of evil-mode) (in evil-mode should see blue cursor, not orange cursor) and
• to be in 'command mode' AKA 'normal mode' (as opposed to 'insert mode') (Esc to switch to command mode) (should see block cursor, not line cursor), and then it's
• u for undo and
• C-r for redo

If you want to redo the last operation, do the following.

1. ESC
2. Do another undo (C + /)

In doom emacs,
Ctrl + Shift + - to undo
Alt + Shift + - to redo