I'm interested in this subject as simple amateur satellite qso don't need big antennas on the roof and still allow for contacts over a much larger area.
The International Space Station is also active in amateur radio. By default it is set up as a 'digipeater' which means it repeats packet radio transmissions. Sometimes there is an Amateur Radio on the ISS (ARISS) school contact where an astronaut answers questions via radio.
These transmissions can easily be received. Use a tracker (see below) to calculate when it will be over your area and receive the signal on any radio capable of tuning to the frequencies of the ISS:
There are also times at which astronauts will answer amateur radio calls
but those are quite rare.
More information at ISS Fan Club and Contact the ISS - ARISS about radio contacts with the ISS.
There are a lot of satellites out there which only transmit signals back to earth. Simple beacons, more complex telemetry information or weather images.
With a possible future exception, all amateur radio satellites are not in a geostationary orbit, meaning you need to do calculations when the satellite will be visible from your location and in what direction.
The easiest is to use a tracking website like Heavens above which has a separate page for Radio amateur satellite passes. Just be sure to set your location correctly.
Sites like this and the tracking software mentioned below use 'orbital elements' (a mathematical description of the orbit the satellite is in) and will want to update them on a regular basis.
I have good experiences with AmsatDroid free. AmsatDroid needs an Internet connection to update the orbital elements, but it can do all the calculations off-line.
I had a look at available tracking software in Ubuntu linux, described at Linux software for predicting satellite passes and doppler shift comparing predict and gpredict.
At the moment gpredict is my choice for real-time doppler shift calculations and predictions. Gpredict wants an Internet connection to update the orbital elements but it can do all the calculations off-line.
The article 145 MHz Moxon by Remco PA3FYM made me think Moxon antennas can be useful for satellite work without building a huge dual-band yagi. I found an article A Simple Fixed Antenna for VHF/UHF Satellite Work by L. B. Cebik, W4RNL (pdf) which shows a usable antenna but the rebuilding instructions are a bit sparse.
But later I decided to put some money into this: I bought an Arrow Antenna Hand Held Portable dual band antenna for satellite work.
I made a cable to use a stereo headphone on my radio so I can listen to the incoming audio with both ears. I hope this helps me in decoding callsigns and other noisy bits.
Video is a great way to demonstrate and explain subjects like this.