A lot depends on what type of CAD you want to do - spatial design or technical drawings.
The coolest software of spatial design is the free SketchUp from Google:
http://sketchup.google.com/
It is particularly orientated towards orthogonal 3D forms such as those of architecture and there are very good tutorials for it:
http://sketchup.google.com/training/
More funky / organic forms can be achieved with some other software. The most professional of these is Blender - an open source project and hence free forever...:
http://www.blender.org/
Also for more intuitive modelling check out the cheap "Moment of Inspiration" at:
http://moi3d.com/
However if what you need are technical drawings the you need software more akin to AutoCAD. You can find an old version of the 2D only AutoCAD LT by Autodesk for under $400 on EBay.
There are free CAD programs come AutoCAD. Here is a good list with reviews:
http://www.eland.org.uk/pages/Misc/cadnotes.html
Some of them blatantly imitate AutoCAD using a shared CAD 'engine' called IntelliCAD. Ther is a long list at the Intellicad.org website but one which caught my eye was the free ProgeCAD:
http://www.progecad.com
However if you just want ot get your head around it then you can dedicate a month to the trial version which you can download from the Autodesk website:
AutoCAD:
http://usa.autodesk.com/adsk/servlet/mform?siteID=123112&id=9106363
AutoCAD LT:
http://usa.autodesk.com/adsk/servlet/mform?siteID=123112&id=10387411
Once you have run through the great tutorials that come with AutoCAD you should be ready to either beg or borrow a copy. Remember that if you are with an educational establishment there are very cheap educational copies availible.
Finally as ever there are a 'torrent' of totally illegal copies floating around the web the use of which I do not condone.
Of course the real value of learning this is probably to advance your job prospects and so once you have work you should be using a legal copy...
All the best,
Mike