For those who availed of 1 month streaming only, You can Download copies (unlimited) of the song but the file is DCF and is only playable via Melon (i think there are other players but it will be on a different blog post). Follow the steps below. ;)
Steps 1-4, Click on the highlighted green bottom of screenshot, Search for the Artist (can be English, but others are in Korean names)
Choose the 3rd one so that you will see the Artist list. (First Tab is their latest albums/singles)
Choose the Artist by clicking on it.
Step 5, Click on the 3rd text (boxed) so you will see the artists' discography.
Choose from the latest to their oldest album by scrolling up or down.
Step 6, once you have chosen, you can click on the Star on the artist, and also give the album a "heart".
Choose the song, or highlight / click all songs in the album if you want to download them all.
Click the download tab, and it will go to another page, where the summary of your download is shown. Click on the Green Button to download, otherwise just hit the back button to go to previous page.
Do Steps 1 to 6 everytime you want to add a new artist's album to your downloaded files. Once done, you want to check your DCF files, right? So follow the next step to do so.
Choose the encircled option to see your DCF List. Then when you go to the next page, click on the first option.
To view DCF files, click on the DCF Tab. You can select all and play all, or Select one by one and then create a playlist of your choice.
Creating Playlist: Choose the bottom option. the first option will only put your songs in the history of "played" items in Melon.
New: Just choose a Name and save.
Rearrange/ Delete Songs: Follow the 2 last screenshots
This will only work if you have wifi or data. The songs will not play if you do not have access to the internet. =)