I got stuck with an iPhone 4. I don't know what music apps I can download with iOS 7.1.2. I want a music app I can make a playlist on

This is really agitating me

You can do that (making playlists) with the stock iOS music app… Of course, you need to either buy your music through iTunes or use the iTunes program on your computer to transfer your music.

You can download music and make playlists on an iPhone 4. That functionality has been around since the very first iPhone. You can do it with the built-in music player.