After securing 3 Instagrams using this method, I think this method probably works the better than most. Pictures will be up to help as soon as I get the chance to make them.
What's Needed:
Internet Connection (seems obvious)
iDevice, and Android Device, or Bluestacks (PC app)
Step 1: Initial Securing
This is in case the old owner immediately tries to take back the Instagram. Go to http://www.Instagram.com/accounts/edit and sign in (if you haven't already). Then change the email and password.
Step 2: Linking Sign into the Instagram you want to swap on your alternate device, then go then upload a picture and where it says share, make sure to share it to Facebook (Twitter might work to, haven't tested it yet), this disallows them from taking back the account.
Step 3: Swapping Note: Be sure not to do this as soon as the Instagram is received, as if the Instagram was revealed to more people than just you more than likely someone is trying to turbo it. Also if you are weird and have a turboer than sign in on that and use that to swap your account
Open up http://www.instagram.com/accounts/edit on your computer and sign into your account you're swapping onto on your alternate device (iPhone, Bluestacks, etc.) and go to Edit Profile. After this change the Username on the account you want to swap and hit save. Immediately after that, change the username to the desired username on your alternate device. If it's available, hit save, if not, go back and try to take the name again.
That's basically all there is to it, like I said in the beginning I will add pictures soon. I also might improve the thread design, but idk.