Using ArcGIS functionality in Python without running ArcMap (e.g. separate editor/environment)?

If I understand the question, you want run your scripts standalone, outside of the ArcMap session. Totally do-able. Save your code into a Python file (.py). As far as editing, you can use IDLE or any text editor (even Notepad) for that matter. Sometimes when you are starting out, it's easier to use a text editor so you can focus on learning the code, not the IDE. Make sure you have your PATH environmental variables setup properly, so they include your Python install directory, something like:

C:\Python26\ArcGIS10.0\Lib\site-packages\;C:\Python26\ArcGIS10.0;C:\Python26\ArcGIS10.0\Scripts

Then it's simply a matter of opening up a command prompt (Start > Run > type 'cmd' [without the quotes]) and typing:

python c:\path\to\python\script.py

Now, this assumes you have all of your data paths hardcoded in the script.

That's the basics. There of course is way more, like using input parameters.

I have just started to get to grips with Python and arcpy and I found PyScripter really useful. Its a full-blown Python IDE

The best way I have found to use Python, is using Eclipse with PyDev. It's a good IDE for developing with and allows for inline debugging, not commend line debugging, and it has allwoed me to build very professional back end Data Management applications on a large scale.

IDLE is only ever that good for testing scriplets, in my opinion, and isn't good for designing complex object based solutions.

It's also incredibly easy to set up.