The document Getting Started With Realistic Neural Modeling suggests several exercises and short projects with GENESIS.
The multi-part hands-on GENESIS Modeling Tutorial contains exercises in most sections.
The links below to subdirectories in Tutorials/exercises give additional exercises that have been used in various modeling courses.
The most recent and complete set of exercises and project suggestions is given in the 2014 GENESIS Exercises collection.
Exercise: Simple GENESIS scripting exercises (Upi Bhalla - 1999 Bangalore course)
Exercise: Create a CA3 pyramidal cell with readcell (traubcell)
Channel modeling, the Traub model, and Neurokit (GENESIS Tutorial 8 - single cell mini-projects)
Exercise: Create a simple two-neuron network (simple-network)
Exercise: The effect of gap junctions on synchronization (gap_junctions)