Nowadays the Internet is usually the fastest way to make subscription changes. At the bottom of many magazine sites you can choose the "Contact" option, and you will usually find a link to Subscriptions and Billing (see Reference as simple example). Whether you do or don't, you can always contact Customer Service at the site. There they will offer an email, phone number or address for further queries. Email is the fastest option, and eliminates possible long hold time on the phone, or need to write a letter, pay for a stamp and mail it.
Making a call is also simple to do. The big difference between calling and emailing is the the usually complicated phone system menu, and the aforementioned possibility of significant wait time. When it comes to get something done more quickly and not having to worry about it when it comes to changing a magazine subscription address, email is the easiest. The phone option, however, may get you in touch with an actual person. Some people find it easier to get their needs across with a live human. And should you end up on hold, you can always put the call on speakerphone and go about you business until your call is answered.
The one advantage to writing a letter for your magazine subscription address changed is that you have a dated physical copy which you can even have registered, if need be, at the post office. A letter offers more peace of mind because it's a physical copy of your request--something you can use to back up your claim. Making a call doesn't have this tangible element, and even email copies can be erased by accident. A mailed and delivery-confirmed letter will take longer but probably offers the greatest assurance. It is direct proof you requested an address change. Post office options for deliveries can be further explored by visiting usps.com.