Designing a simple CW transmitter/receiver
To actually use a CW transmitter in the ham bands you would need a license. Further you need to insure that you do not cause harmful interference to other users which could happen because of a problem with the circuit.

Why not start with a receiver which does not require a license? You could start with a simple crystal set. Here is a YouTube video showing how to make one: http://www.youtube.com/watch?v=skKmwT0EccE

You will learn skills like how to wind a coil, how to tinker with a debug a simple circuit to get it working etc. It will not require much time or money.

Once you get it working you can re-use the parts to make a more sophisticated one transistor set. More importantly you will be reusing the skills you acquired getting the first set to work.