Premier Elements doesn't automatically sync sound, but it's easy enough to do by matching the sound waveforms from the camera and recorder. You can help yourself by imitating the clapper that the pros use - just clap your hands either at the beginning or end of the clip. You'll see it on the waveform tracks in Premier. If your camera is not recording sound, you'll match the sound of the clap to the hands meeting on the video - just like old-time Hollywood editors before the days of timecode.

Rode makes an inexpensive shotgun mic for video that will mount on the hot shoe of your DSLR, if you're using that for video. It records good sound.