Looks like I'm gonna be the lone pissant and state a dissenting opinion.
This is a flat latch Chief's Special. It's an old gun, and a collectible one. Refinishing it makes it into a pariah among collectors. It was someone's carry gun. The marks on it are it's history. If you refinish it, you just removed that history. You cannot put that history back. It's gone. Don't refinish it.
If you want a pretty new gun, go buy one from S&W. If you want a pretty new blue gun, buy a Taurus. Heck, I'll straight up trade you a shiny blue Chief's Special for your old flat latch beater, even Steven, if you want a old Model 36. Don't destroy history.
The grips offered on S&W's webpage are not appropriate for this revolver. They will fit, but they are not appropriate. It should be wearing diamond walnut grips. You can find them on ebay and occasionally at gunshows.
Finally, if possible, post a good photo of this gun. Both sides. The so called flat latch actually came in three variations. Give us the serial with the last two digits Xed out. It's on the butt. Find out what you have before you reblue it. Then decide whether you want to add to it's history or erase it forever. You have all the time in the world to reblue it, but once it's reblued, you can never restore the original finish.
In the end though, it's your gun. Do with it what you will. Just make an informed decision.