The command that you are looking for is Ripple Delete from the Edit menu and there are Keyboard shortcuts for the various options. Rather than just Deleting the selected bit of audio in a clip and leaving a gap this command deletes the selected audio and moves up the following audio. See the Help files for more details of the options.
If you have already chopped bits out and have gaps left then:
"Collapse a gap between clips on a track
- Right-click the empty area between the clips, and choose Ripple Delete > Gap."