Start by separating the outside from the inside. The case, crystal and chain can be cleaned gently by an owner. The movement cannot. A movement needs to be taken apart, cleaned in dedicated solutions and oiled again with the correct oils. Only a watchmaker can do that job.
Case: wipe it with a soft, dry microfibre cloth. On gold or silver, a jeweller's cloth can remove light tarnish, but polish very sparingly. Every polish takes off metal and softens hallmarks, engraving and case edges. Gold-filled and plated cases can wear through to the base metal. Never use abrasive pastes or ultrasonic jewellery cleaners on a complete watch.
Crystal: glass can be wiped with a slightly damp cloth. Acrylic (plastic) crystals scratch easily, and a specialist polish can remove light marks. Do not let liquid reach the gap around the crystal bezel.
Dial: do not clean it. Enamel dials can take a very light dry dusting at most. Painted, silvered and printed metal dials can lose their markings as soon as they are touched with a damp cloth or solvent.
Movement: do not spray, oil or wash it. WD-40, lighter fluid and household oils leave residue that attracts dirt and damages pivots. If the watch has stopped or runs erratically, it needs a professional service, not a clean.
Chains: most metal chains can be cleaned with a soft brush and mild soapy water, then dried thoroughly. Keep this well away from the watch itself.