Here is a first-rate video editing tutorial that shows how with a simple little editing technique you can smooth out your audio and video editing.

The “J” cut involves hearing a scene before you see it.

The “L” cut involves covering part of a scene with the next scene while maintaining the previous scene’s audio.

Let’s be careful though that you do not create ethical problems along the way.  For example if you are showing scenes of a playground, then roll in police sirens, you are implying that you could hear the sirens at the playground.  Is it true or were they on opposite sides of town?