Text Blocks
Each shape in Visio has an associated text area associated with it. This text area or text block will always move with the shape.
To move the text block for a shape:
Click on the arrow which is just to the right of the text tool (The text tool is the letter A which looks like the bold button)
Select the Text Block Tool. You can then move the text block -or- click on the pointer tool to move the whole shape.
Get in the habit of switching back to the text tool when you have finished moving text blocks.