David Schneider built his own radio telescope out of roof flashing, an empty paint thinner can, a free software-defined radio app, USB receiver, and a length of coaxial cable. The whole project cost him less than $150 and he's already used it to detect galactic hydrogen and monitor the motion of our Milky Way galaxy's spiral arms. (With a radio telescope, you look for and measure radio-frequency radiation emitted by astronomical objects.) From IEEE Spectrum:
Point at Cygnus and you'll receive a strong signal from the local arm of the Milky Way very near the expected 1420.4-MHz frequency. Point it toward Cassiopeia, at a higher galactic longitude, and you'll see the hydrogen-line signal shift to 1420.5 MHz—a subtle Doppler shift indicating that the material giving off these radio waves is speeding toward us in a relative sense. With some hunting, you may be able to discern two or more distinct signals at different frequencies coming from different spiral arms of the Milky Way.
Don't expect to hear E.T., but being able to map the Milky Way in this fashion feels strangely empowering. It'll be $150 well spent.