Are the files in your iTunes library? If they are, you can convert the m4p files to mp3 files if you imported them from a cd. If you downloaded the songs from iTunes, you may not be able to convert the songs.
*To convert a song's file format:
Choose iTunes > Preferences, then click the Advanced button at the top of the window and click Importing.
From the Import Using pop-up menu, choose the encoding format that you want to convert the song to, then click OK to save the settings.
Select one or more songs in your library, then choose Advanced > Convert Selection to MP3, Convert Selection to AAC, Convert Selection to Apple Lossless, Convert Selection to AIFF, or Convert Selection to WAV. (The menu item changes to show what's selected in your Importing preferences.)
To convert all the songs in a folder or on a disk, hold down the Option key and choose Advanced > "Convert Selection to," then choose the folder or disk containing the songs you want to convert. All the songs in the folder or on the disk will be converted except songs you purchased from the iTunes Store. (Purchased songs are encoded using a protected AAC format that prevents them from being converted.)
*Reference from iTunes help
If you're able to convert your songs into mp3 format, all you have to do is drag them into your mp3 player. I use a 3rd party mp3 player also, and I had to do this for most of my music before I could get the songs to appear in the player.