It is relatively easy to cancel magazine subscriptions on iPhone or iPad. The process remains same across iOS 7 and the older, iOS 6. If you have subscribed to magazines that are set to auto-renew, the kill switch is the very auto-renewal toggle.
Your subscriptions are not controlled through the Newsstand or the Settings app. Instead, since they are linked to the Apple ID via App Store, you control the subscriptions through the Apple ID settings.
Step 1: Open the App Store
Step 2. In the Featured tab, scroll down to the very bottom and tap on your Apple ID. (you might be asked to login to your Apple ID)
Step 3. In the popup, tap on View Apple ID
Step 4. You are taken to the Account Settings page. Do you see Subscriptions?
Step 5. If there’s no Subscriptions option (with a Manage button), it means your Apple ID has no Newsstand/magazine subscriptions.
Step 6. If there is, tap on the Manage button
Step 7. This will list all your magazine subscriptions. Tap the one you want to cancel.
Step 8. When you tap on the subscription, under Renewal Options there’s an Auto-renewal toggle. Switch this off.
After the period of the last renewal is up, your subscription will be canceled automatically. There is no other kill-switch to cancel a subscription. You just stop the auto-renewal and the subscription runs out.
As you can probably see, you can also manage your subscription from here. You can change the frequency of the auto-renewal of any subscription just by tapping on the options. Typically, most subscriptions let you go for a weekly or an yearly subscription.
Note that deleting a magazine from your Newsstand gets rid of the magazine and the data it stores on your iPhone/iPad. But the subscription is something you need to cancel through the Apple ID account settings.