The simplest one is the second option you asked for.

When you have finished creating the form, use the 'Save as...' option, and in the 'Save as type...' drop-down box (below 'File name...') and choose 'Document Template'

NOTE 1: be careful - when you choose the Template option - it changes the save location to the Microsoft Office Template Default location, so you will need to change it again.

NOTE 2: it's best if you save the Template to a location where most users don't know it is saved, and then put a link to it on the user's desktop. The reason for this, is that if you double-click a Template, it will open a new document (forcing a rename), but if you right-click and choose 'Open', you open the Template for editing! All it takes is for one person to do this to the original and you are right back at the beginning again!!

NOTE 3: It IS possible to make the document auto name itself by using a name in a given field - this would need to be written in VBA script though! Pro: auto names itself - Con: when you go to open it, it comes up with a warning message that, if 'No' is clicked, will disable the autonaming! Personally I would only suggest this option if: 1) you know how to create a 'Trusted Folder' for your network (thus preventing the warning from coming up), or 2) if ALL the people that are going to be using this document understand the basics of macros and macro security.
If you wish to have help creating the script, let me know and I'll give you as much or as little help as you want.