"Geistig" is everything concerning your mental skills, everything related to your brain, intellectual things, so to speak.
e.g. geistig behindert - mentally handicapped
you could do math as a mental workout to exercise your brain. That would be "geistige Arbeit".
The humanities (philosophy...) are called "Geistes-wissenschaften".
"Geistlich" is related to church and faith; religion. A "Geistlicher" is a man of God, for example a priest.
I just looked up the difference between the suffixes because as a native speaker I had never thought about it before. What I found is this:
-ig always tells us that whatever the root is, is present at the moment.
e.g. nebelig (there is fog), sonnig (there is sun)
-isch tells us how something/someone is.
e.g. neidisch (from "Neid"), russisch (from "Russe")
-lich is pretty neutral and only means that something has to do with whatever the root is.
e.g. everything that's "sommerlich" has something to do with "Sommer", everything that's "weihnachtlich" has got something to do with "Weihnachten"
hope that helps!