Savage,
Your $425 is well spent. You have a Smith and Wesson model 19. Probably a 19-5 to be specific based on the date given when someone researched your serial number. This model is one of the best, and most successful guns Smith and Wesson has ever made.
The model 19 is built on Smith and Wessons K-frame. The k-Frame has been in existence for over 100 years and was mainly used to chamber .38 Smith&Wesson, and then .38 special.
In the mid 1950s a well respected gun expert and border patrol agent named Bill Jordan convinced Smith and Wesson to build a .357 magnum in the K-frame. He felt it would be the ideal combination of size and firepower for Police forces. More on .357 loads for this gun a bit later on.
You have been given a lot of advice in this thread, most of it good, but a lot of it contradictory.
If you will indulge me, let me offer what I would do and the order I would do it in.
Get a set of replacement stocks. We refer to Smith and Wesson grips as stocks. It's an old term, when they made revolver grips out of wood, like the real nice stocks for rifles. So if you are going to have a Smith and Wesson in your collection, we need to welcome you to our fraternity, and teach you the lingo. Because you have a 6" barrel, your model 19 has what is referred to as a "square butt" grip frame. So if you get new stocks, you will want to make sure you get K frame Square Butt stocks. Do not get Round to Square conversion stocks. They won't fit your gun.
By the way, if you want the stocks restored, there is a member on the boards that does amazing work fixing these stocks. Yours will clean up very nicely. I will get you that information, and add it to this thread.
Ok so once you have some new stocks, give the revolver a good cleaning and some light lubrication. Clean the barrel and all 6 charging holes on the cylinder. Use a good cleaning kit with brass brushes, good cleaner like Hoppes, and run plenty of patches through the barrel and charging holes.
Get a box of .38 special ammunition, and a box of 158 grain .357 magnum ammunition. You DO NOT want to shoot light .357s in this gun. 125 grain .357s are a great round and common. Chevy will wear this gun out. In general, don't shot a lot of .357s in this gun, but really stay away from 125 grain loads as they have been known to crack forcing cones.
So shot it, and get a feel for it. The trigger should be smooth in single action and double action.
See how you like it.
Then if you decide it needs to be cleaned up, I would oil and wipe it repeatedly and see how much of the freckling you can get off of it.
But you will love the gun.
By the way, on YouTube, there is a great gun channel called hickock45, he has some videos on the model 19.
Check these out:
https://m.youtube.com/watch?v=mLhiDlcktdY
https://m.youtube.com/watch?v=K0oFg3O-DL4
https://m.youtube.com/watch?v=FBPu2qbhiBA
And here is one about the model 19s Stainless Brother, the Model 66
https://m.youtube.com/watch?v=S-UF6CpThdk
Hickock shows what they can do, and gives a nice history lesson.
Welcome to the Smith and Wesson brotherhood, they get in your blood.