A receiver is an excellent first project—it can allow you to experience the magic of shortwave before you get your ham ticket. 

The MRX-40 Mini Receiver 
QST September 1997, pp. 59-60
A tiny 40 meter amateur band Morse code receiver barely larger than a half dollar.

Rescaling the MRX-40 Receiver for 80 Meters
QST May 2001, pp. 98-99 and the July 2001 p.31 Feedback

The Neophyte Receiver 
QST February 1988 pp. 14-18
A simple 80 or 40 meter amateur band AM, Morse Code (CW) and Single Side Band (SSB) receiver.

A Binaural IQ Receiver by Rick Campbell, KK7B. This experimental receiver takes advantage of our sophisticated stereo hearing system. QST March 1999 pp.44-48. 

High-Performance, Single-Signal Direct-Conversion Receivers describes the "R2" by Rick Cambell KK7B.  January 1993 p.32-40.

 HBR Receiver Web Site

These are the tube type HBR-14, HBR-16, HBR-8, HBR-11, HBR-12, and HBR-13C receivers originally published in QST. Parts, and sometimes entire receivers, can be found at swapfests.

Home-Brew Regenerative Receiver.  A homebrew tube receiver using a 6V6 detector and a  6SN7 audio amplifier.