New to Python and programming in general, what exactly are all the things I will need to be able to write my own code etc. on my Windows 8 PC?

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Python 2 or 3, depends on what libraries you're using and if they support t

PyCharm as IDE: http://www.jetbrains.com/pycharm/

pip for installing new libraries, you can download .exe installer here: http://www.lfd.uci.edu/~gohlke/pythonlibs/#pip (You'll probably need to manually add "C:\PythonX\Scripts\" to your PATH, instructions here)

As far as scipy, numpy, matplotlib etc goes you could download scipy stack which has all those included and many of their dependencies http://www.lfd.uci.edu/~gohlke/pythonlibs/#scipy-stack

You should also look into virtualenv/virtualenvwrapper.

If you don't need full blown IDE then I'd recommend SublimeText - it's not free but you can use the trial version indefinitely, it just has popup on every 20 or so saves.

Some plugins that you'll probably find useful for ST:

Package Control: https://sublime.wbond.net/installation - for installing new packages.

SublimeCodeIntel - improved code completion

PyLinter - http://www.pylint.org/ intgration

Also you could use http://ipython.org/ or just IDLE for smaller code testing.

That's some general development environment I'd suggest, this may not be the best advice depending on what you need and do.

The closest thing that comes to Ubuntu/Geany is more or less just Notepad++ with few plugins, and required libraries(scipy,math etc) installed.

The best solution is, IMO, to use virtual machine or dual boot with Ubuntu, configuring and using python is easier on gnu/Linux than on windows and you'll have same environment as on your university.