I am using Premiere Elements 15 on a PC, and I'm having trouble deleting gaps where I've trimmed my video. None of the other forums or video tutorials have helped.

I'm using PrE 15 on a PC and don't have trouble with gaps.  So, it would help if you explain how the gaps are getting there. 

Otherwise, this is from the manual:

You can quickly delete empty space between clips in the Expert view timeline by using the Delete And Close Gap command. Alternatively, press the Delete or Backspace key. Both techniques shift adjacent clips over to fill the gap. In the Expert view timeline, do one of the following:

•Right-click the empty space, and choose Delete And Close Gap.

•Select the space you want to delete, and press the Delete or Backspace key.

note: If the gap is small and difficult to select, move the current-time indicator to the gap and click the Zoom In button.