My MP3 player won't play MP4 files.

You can't convert iTMS tracks in iTunes as they are copy protected. If you need them in MP3 format you can burn them to audio CD in Tunes and import them back as MP3:
Burning an audio CD in iTunes for WindowsHow To Choose Import Settings

The songs sold in the iTunes music store are not mp3s, they are protected AACs (i.e. "mp4's"). They will not directly play in your player. To get them into mp3, use iTunes to burn the purchased tracks to an audio CD, and then import the tracks from the CD using iTunes. Before you import, make sure your preferences have your import format set to mp3.