First, open up melon:
https://www.melon.com/

On the top right over the ID/password entry you should see the text "회원가입" - click there.

Select the center option, "일반회원가입" - join as a regular member.

On the next screen, click to agree to the terms and conditions (there are several check boxes), then "agree" at the bottom right.

Next, fill in your member information.

Check that your ID is available by clicking the button at the right.
You should get this message: 사용할 수 있는 아이디입니다.(This ID is usable)
Then you will have a green check mark which shows that ID is good to go.

Your password should include letters and numbers.
Your name should be written in lower case (abcde) with nospacesbetweenthewords.
When you are finished filling everything in, click "NEXT" at the bottom.

※ If you want to link this account with your nate account, you can do that by using your SK Planet One ID (go to the end of this document)

On the next screen, you can verify your account by phone SMS (you will type in the code number that you get in the text message), or by email which is probably easier.
I'm sorry, I forgot to take a screen capture, but it looks almost exactly like this screen for nate. The picture shows the email option (the second tab, 이메일인증)

This is the next screen you will get, which basically just says that they have sent the verification email.

Check your email, which should come from "meloninfo", and click on the "complete signing-up" button.

Your registration is now complete!
You should be redirected to this screen on melon, which says your email registration is complete and after you click the "confirm" button at the bottom you can start using member services.

Now try logging into melon with the One ID and password you chose.
You cannot purchase or stream on melon without a Korean ID, but being able to log in gives you access to voting, commenting, "liking" songs and other benefits.
You can vote for Teen Top on Show Music Champion, too! See here.

※ If you wish to link your melon account to your nate account, you can do so because they both are connected to the same "SK Planet One" service. To do this, you will enter the same SK Planet One ID on the screen where you choose your username for melon, etc.

This is a little trickier and I had to click around a little, but it should be pretty much these steps.

Enter the same One ID you used for nate/cyworld sign-up and when you click "check availability" you should get this pop-up window. To confirm the tie-up you need to enter the One ID again or the email (I used the email) and click the green button.

Your email address should be filled in automatically on the next page, just click the green "Complete" button at the bottom to send out the mail.

Now check your email and you should have a message from melon. Click on the "complete signing-up" button at the bottom.
(Re-using the pics from above for the next two steps because they look exactly the same)

You should be redirected to this screen on melon, which says your email registration is complete and after you click the "confirm" button at the bottom you can start using member services.

Now go to log-in on melon.
The first time I tried to log in to melon using the test account I made which was linked to the One ID I created for nate, I got this pop-up which said I had to agree to the terms and conditions:

(I think there was another screen where I did so, sorry)
After that I got this screen which says terms and conditions agreement is complete, and only then was I able to log in. Putting these screen caps in just in case!