Press [ on your keyboard when playing to set the start of the selection. Press ] on your keyboard when you hear the point for the end of the selection, which will then create the selection.
You can then use Cut (which removes the selection) or Trim (which removes everything except the selection) according to what you want to do. Trim and Cut both have buttons on Edit Toolbar (click the link to read about Edit Toolbar): http://manual.audacityteam.org/o/man/edit_toolbar.html .
If you have hidden Edit Toolbar, choose View > Toolbars > Edit Toolbar.
The only feature requests here that Audacity does not yet have would be:
Have buttons for the [ and ] keyboard shortcuts
Have [ and ] actually set markers in the waveform.
If you want to vote for either of those two additional features, please let us know.
and change the keyboard bindings for “Left at Playback Position” and “Right at Playback Position” to a key you can press without using a modifier. You can choose anything that does not conflict with an existing shortcut.
So to explain how to make the change once you have the Keyboard Preferences open:
In the “Category” drop-down, choose “Edit”
Half way down the list, find “Left at Playback Position” and click on it to select it
Click in the text box underneath the list then type the key you want to be the new shortcut for “Left at Playback Position”
Repeat steps 1 to 4 for “Right at Playback Position” which is underneath “Left at Playback Position” in the list