How can I ditch iTunes and find a suitable audio player?

I suggested you try SongBird.

also you can choose:

• PicoPlay
• VoxApp
• Banshee

After doing a Google quest for a free audio player I found Instinctiv.
It has a great UI and Mac look and calls itself "A Better Music Experience". Is anyone aware of this application? How do you feel about it?

After a little test drive with the software I can say that it really is an enjoyable experience and the announced smart shuffle is a nice feature. It uses the iTunes library so you can just press play and you'll be fine.

Vox (free with in-app purchase)

A great simple application for music playing. I love this app. It's sooo simple to use and to use. It also integrates nicely with your iTunes library (if you so desire).

UPDATE: Everplay is no longer developed (and nowhere to be found).

Leaving the answer around in case someone eventually takes over. Also there are alternatives to Spotify these days, like Google All Access Music, iTunes Match, Amazon Music, etc. As the world moves towards cloud based alternatives, you probably want to consider these too.

Everplay is a nice iTunes alternative because it can read the iTunes library but it’s way more light, given that it doesn’t attempt to cook breakfast and wash your car. It’s a music player.

It’s not free but offers a nice trial. Give it a shot.

Of course, if you are in Europe, you can always use the excellent (and free) Spotify.