Chain - Use the “12 links equals 12 inches” rule. Measurements of 12 1/8”+ are grounds for replacement. Use a chain wear indicators for precise measurement. A worn chain will quickly wear your chainrings and cassettes.
Chainrings/Cogs - Inspect for straightness, bad teeth or small chips. Excessive chain suck with a new chain usually indicates that you need a new chainring. Check chainring bolts for tightness using a torque wrench for accuracy. Do not overtighten.
Crank/Bottom Bracket - Check for looseness/smoothness. Remove the chain and spin the cranks around. They should spin freely. If tightening crank bolts doesn’t solve the problem, rework bottom bracket (cartridge or cup-and-seal).
Handlebars - Check for bending or looseness. If it’s bent or if there are any indications of cracks or stress areas, replace it.
Pedals - Check for axle play by wiggling the pedal. Clean and repack bearings once a year at least. Tighten clips and check straps/clips for excessive wear. Clipless folks check release entities for wear/lubrication.
Saddle/Seatpost - Look for saddle rail deformities or cuts in the upholstered material. Lube seatpost liberally with white lithium grease for proper operation. Use electrical tape on small tears.
Wheels - Check rims for cracks, dings, dents, loose or broken spokes or another deformity. True your wheels. If you can’t do it yourself, take it to a shop trued wheel is a strong wheel. Check hub bearings for smoothness/lack of play.