What should I be doing and how often?
Depends on the bike. Do you have pivot grease ports?
Was clean lube the moving bits and keep ridong.
Make sure you check bolts to see they are tight
How sensitive are you to changes? I don't check bolts as they don't often come loose and I notice when/if they do. YMMV.
I just look after the shock as I do the forks and watch for anything sticking, rattling or knocking.
Squirt some silicon lube on shock seals? A dribble of proper oil into pivot bolt/bearings now and again if I remember?
But mostly just brush off any mud and never aim the jet wash at pivots or bearings. Treat like any other bike.
I've had bolts come loose on my Trek. However they've conveniently had torque settings laser etched on the caps - thanks Trek!
Other than that I just spray Fork Juice on the shock shaft but it probably doesn't need it. Nothing to do really.