How do I set up my iPod Touch to play all songs?

It will take a bit of detective work to find out what is causing that.

It could be:

• there has been an ongoing issue for a long time now whereby songs do not get added to the iPod at Sync time. It looks as though they have been, but instead, they are greyed out: the player lists them but cannot play them. Yo can test for this; if you can identify a song that didn't play when you expected it to, can you manually select it to play it or does it get skipped?
• if you have the player on shuffle, any songs set (in your iTunes library) to Skip when shuffling will not be played. Again, you can test for this. If you can identify a song that didn't play in shuffle mode when you expected it to (i.e. the player plays everything else in the playlist and then stops), check the song in your iTunes Library. Select it and click on Edit/Song Info/Options>Skip when shuffling. If there's a tick in the box, the song is set to skip when shuffling. You can turn it off and then Sync the iPod with the library and that will correct the issue
• in theory, if you put the player into pause mode, it should start from where you stopped it and continue the playlist. However, if you if you stop the player completely, so that the Now Playing ribbon says Nothing Playing, or you turn the iPod off (rather than putting it into sleep mode), then when you re-start it, it will treat the playlist as being re-started too, hence playing the same songs. It's possible that the music player may forget the play position if you use the Podcast app.