The method recommended by Instagram itself is the Instaport service. Now you can download all of your files in a convenient zip folder by simply linking your account. In the future they will allow an easier way to port your pictures to Facebook, or Flickr, but for now you will have to upload your photos manually. You can also download your tagged and favorite photos from friends. After you know that your data is stored elsewhere on a reliable server, you can then delete your account without having to worry.
There are a few other options out there, so let us have a look at them.
This service is really good for syncing your photos to storage such as dropbox or Amazon and is far more flexible. By linking your account you can download pictures directly to your computer, or port them to whatever service you prefer.
While it does not yet have a way to port your photos directly from Instagram, you can have your pictures uploaded to their servers in the background while you go about your daily business online. First though, you have to make sure all the pictures you want are on your hard drive, so it takes an extra step. Right now, storage of photos is free as long as you let Google convert your images to fit within their maximum size requirements ( 2048x2048). If you would like to preserve your higher resolution photos they will let you store up to 5GB for free in Google drive. Simply select the option to upload full size pictures and they will go directly to your drive.
Have no fears of privacy breaches as all uploads are set to private by default. This means that no one can see your albums until you give them express permission by either a link, or sharing through Google+
I hope this helps anyone scrambling for a way to get out of Instagram before they end up on a billboard somewhere.