for decision, which antenna to use, find answers to the questions:

  - which frequency should be used?
  - omni or directional antenna? radiation pattern: pancake, lobe etc?
  - mounting and drive (az/el rotor and control) ?
  - polarisation of the signals (linear or circular?)
  - required / desired / achievable gain

for amateur radio satellites you would answer these questions e.g.:

  - 430-440 MHz
  - directional antenna with 3.) azimuth and elevation control
  - circular polarisation
  - 12 to 18 dB is required and can be achieved with 1 to 4  20-el X-Yagis

with such antennas you can transmit to the existing hamsats
with only 10-50 Watt rf power and you get a good receiver signal

Antenna construction is a very wide field,
search the net, especially for ham-radio antennas.
Homemadekeing antennas is a nice and low cost occupation.
There exists also free simulation software for antennas.
Don't ask where exactly, just Google for keywords:
Antenna +   yagi, beam, turnstyle, lindenblad, elevation,
azimuth, circular, polarisation, reflector, dish, patch,
stacked array .
e.g. http://www.amsat.org/amsat/articles/w6shp/lindy.html