I want to select a portion of audio and then move it by dragging it to different areas in the same track.
How do I accomplish this?
Yes, you can move tracks around without cutting and pasting!
At the top of the Audacity screen, just right of the transport buttons there are several "Tool" buttons. Click this button:
This is called the Timeshift tool. Use the Timeshift tool to move tracks along the timeline. After selecting the Timeshift tool, select a track by clicking on the waveform and dragging it left or right to move it earlier or later in the timeline. Or you can click on the waveform and move it to any other track anywhere on the timeline.
Credit to wikieducator.org/Using_Audacity/Moving_tracks
There is no way to move audio clips around via drag-'n'-drop.
In case someone is wondering about this while using recent versions, time shift functionality mentioned in this answer was removed and replaced with this barely visible draggable part near the top: