AmSat and other Satellites
The place to start here is the AmSat home page.
There are a number of Amateur Radio satellites orbiting our planet, plus a number of others with Amateur Radio usable equipment. There are other satellites of interest, including the weather (WX) satellites. It doesn't take much to start using the LEO (Low Earth Orbit) satellites, even a hand-held can be enough … a number of people have had QSOs with the ISS using just a hand-held.
There are a variety of antenna systems that can be employed to improve satellite access, including a number of omni-directional designs, such as a mag mount with whip, a ground plane antenna, a turnstile antenna, an egg beater or Eggbeater II, and the EZ-Lindenblad. The ultimate antenna systems use multi-element yagis and AZ-EL (Azimuth-Elevation) rotators.
There is also an AmSat e-mail list called "amsat-bb" that can be subscribed to via the AmSat web site.