1. Remove shift cables. Dissemble levers, clean, inspect, lube and reassemble. Lube rear derailleur pivots and ensure it moves freely. Install new cables and adjust (obtain from LBS).
2. Remove brake cables. Clean and lube levers and check for free movement. Lube caliper pivot and ensure it moves freely. Install new brake cables (obtain from LBS). Remove brake shoes and take them to LBS to find appropriate replacements.
3. Any general purpose grease is appropriate for bearings. Clean out old grease with solvent first. For the chain, get chain lube (not grease) at the LBS.

(LBS = Local Bike Shop)

(I wast writing this as you were posting your pix so hadn't seen them)