Is it a geared machine or the fixed? I assume it's a "leave at the station" machine so probably not one I've seen.
Inspect tyres, look for cuts, perishing, bulges, are they straight? (if not is it the wheel, the tyre or just the fit of the tyre). Try pumping up to working pressure plus a bit and recheck, then deflate to normal working pressure.
Check chain - stretch/wear using a gauge and inspect (missing rollers, rotten bits, sticky bits) static and pedalling. If badly stretched may need a new cassette/freewheel/sprocket as well as chain.
Feel of BB and pedals (tightness or play are the warning signs) - all Shimano pedals I know of have adjustable cup and cone bearings if there is play in them, BB's are mostly throw away items.
Rear mech alignment, play or tightness in rear mech pivots and jockey wheels.
Headset/steering - check/adjust for free movement with no play and no "indexing".
Do the bounce test - if anything rattles either sort it out or live with it.
Bar tape if drop bars.
Check saddle height and reach, may need tweaking after a test ride.
I can recommend some very good hand cleaner (citrus flavoured). Wash hands or remove filthy gloves between doing anything to the dirty bits (brakes or transmission) and checking at the handlebar end.
The tooling arsenal is available if you need it, you know where I live.