How to listen to music samples with new Apple Music?

Where are you searching for them? If you are in the Browse or "For You" sections in iTunes, then that requires a subscription to play.

Make sure "Store" is selected instead, and search for them there.

What does “new Apple Music” mean in your title?

If you subscribe to Apple Music, you may listen to whatever you like, but I cannot imagine what “without iTunes” means, as of course you have iTunes on your Mac.