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If which python, when run by the normal user, returns ~/bin/python, then this is a different Python from the one run by the superuser, which, according to what you say, is /bin/python. sudo pip install django runs as the superuser and installs Django in the system’s Python environment. When, afterwards, you execute python manage.py shell, it runs another Python, and that other Python’s environment apparently doesn’t have Django installed.
If you run pip install django without the sudo it might work, but since you have superuser permission on that machine it would be better to get rid of the Python installation that you have in the normal user’s home directory, and make sure your system has only one Python installation until you really know what you are doing. Otherwise you will be confused whether you use virtualenv or not.
Edit: I disagree with many comments that tell you to use virtualenv. You are confused enough without it. Let’s simplify the problem first. Remove all virtualenvs you have created (you do this by deleting the directories) and forget everything about virtualenv. Logout and login again (this will ensure the virtualenvs are deactivated). Don’t modify your manage.py, it’s fine as it was originally. Install Django system-wide with sudo pip install django, then try to run your Django project with python manage.py shell. Only after you get it working and you start having a grip on the system go on to start playing with virtualenv. Virtualenv is great, but everything in its time. (virtualenv demystified is an introduction to virtualenv written by me.)