How do I move audio from one portion to another in Audacity?

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

1. Select the portion of audio you want to move with the mouse
2. Cut the selection (either with the keyboard - Ctrl+X on Windows/Linux and Cmd+X on Mac - or with the mouse - click on the Cut icon in the toolbar or go to Edit → Cut)
3. Select the place you want to move it to with the mouse.
4. Paste the selection (either with the keyboard - Ctrl+V on Windows/Linux and Cmd+V on Mac - or with the mouse - click on the Cut icon in the toolbar or go to Edit → Paste)

There is no way to move audio clips around via drag-'n'-drop.