I have a Kindle Fire HD and also have the Kindle reading app on my PC. I can view my Kindle books on my PC, but not magazines. I also cannot transfer a current subscription to a Kindle Fire HD magazine to my PC because the subscription itself is only available in Kindle Fire, Kindle Fire HD or Kindle for iPad. In the "manage my kindle" section of Amazon's website, the options to transfer to other than my Kindle Fire is grayed out.
This is from Amazon: This publication is not available for some devices. The publisher may have opted out of making it available on certain devices, or the reading experience may not yet be optimized for this publication on those devices.
Something to think about if you're planning on getting a Kindle Fire magazine subscription.