It is a good idea to make a visual record of your researches with examples, notes and trials. You might wish to begin your study of calligraphy by making a visual record (use photocopies or print outs from the web) of calligraphic writing from different cultures. See if you can relate the work not only by culture but by the types of writing tools used. Also note the date of the original artwork. This collection could take the form of an ongoing record of visual references.
Insight into the differences between writing systems can be gained by researching current theories on the origin of writing. Your group might find it useful to break this topic down into parts. Writing seems to have arisen independently in various parts of the world. Areas for research would include the origins of writing in Asia (many possibilities for sub division here also), the Middle East and Central America. Each group could make a short presentation to the class of their findings. It is always worth checking the web for articles on this subject as new discoveries are being made all the time and such articles make good discussion material for seminars.
Writing systems vary from culture to culture. So you will find it helpful to know the difference between a system based on an alphabet (the system used to write English for instance) and an Ideographic system (used for Chinese). You may like to make notes towards your own definitions of these terms. Other phrases to research include, the rebus principal, determinatives.