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.
Exercise: Simple GENESIS scripting exercises (tutorial2)
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)