3D CAD design software is available in a number of shapes and sizes. Most of the packages mentioned below offer fully functioning trial versions for 15-30 days, so definitely try before you buy.
At the top end, costing tens of thousands of dollars are programs like Catia, which will allow design and simulation of assemblies with hundreds of parts. This type of software is typically found in large corporations, e.g. automakers.
In the middle, in the $2k to $10k range are programs like Solidworks (the market leader), Solid Edge (easier to use and aggressively challenging Solidworks), SpaceClaim (very capable, aimed at 'engineers' instead of draftsmen), Key Creator (similar to SpaceClaim), AutoDesk Inventor and PTC's Creo.
(PTC also has a free program Elements/Direct Modeling Express which is quite capable and highly recommended for an introduction to serious direct modeling. Here's a 30 minute Direct Modeling Express tutorial to get you started.)
At the lower price end ($200-$2000)are programs like Alibre (similar core functionalities as Solidworks) and TurboCad. Rhinoceros is a NURBS modeling program that some find more intuitive to work with. Plus you can find it for <$1000 (make sure to shop around).
It's also possible to get started with 3D CAD for free with Google Sketchup. It has limited capabilites and you will need a third party plugin (e.g. this) to export in .stl format, but again, it's free.
Finally, if you're adventurous and prefer command line to GUI, the open source OpenSCAD might be for you. A user on Thingiverse has even uploaded a guide and files for using Notepad++ as your OpenSCAD text editor.