Trying to put a value on anything with just the limited information provided is hard to do. Value depends on a combination of many factors. You have only provided the most generic description. Without a detailed description to include, finish (is it factory or aftemarket), barrel length, overall condition, stock material, sights, box, original paperwork and any accessories, only a general estimate can be given. This weapon could be in the 50-350 USD price range.
100-1000 USD
You have a Smith and Wesson model 14 38 special. The 38 refers to the diameter of the barrel. That barrel is 38/100 of an inch in diameter. A 45 automatic has a barrel that is 45/100 of an inch in diameter. A 9mm has a barrel with a 9mm diameter. You can shoot 9mm ammunition from a 38 special. A 22 pistol has a small diameter barrel.
400 or so
Age of smith Wesson serial no 5K4166
100-300 USD
ctg = cartridge
50-500 usd
Depends on exact model, barrel, and condition. Around $200,
100-1000 USD
100-400 or so
50- 450 usd
An experienced gunsmith can or you can send it to S&W to have it done.