for decision, which antenna to use, find answers to the questions:
for amateur radio satellites you would answer these questions e.g.:
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