Yes, you have to download them first to your PC, then to your Zune via the Zune software. If you're trying to save space on your PC the good news is that once a song is on your Zune, you can get rid of it from your collection. And it will remain on the player. If you want to get it back, you can re-download it from the store at a later date.

Second, you can either sync everything, or selectively choose what you want to sync to the device by right clicking on the content and choosing "sync to (ZUNE NAME)" from the right click options menu.