prorope:
You have a S&W .38 Safety Hammerless 3rd model manufactured between 1890 & 1898 in the serial range 42484 to 116002. Caliber is .38 S&W, not .38 Short Colt or .38 S&W Special.
The Safety Hammerless was manufactured in .32 S&W & .38 S&W, they were also known as the New Departure (due to the new design) and unofficially as the 'Lemon Squeezer' due to the grip safety.
Your gun was designed to shoot black powder loads, I would stick to these or an equivalent black powder substitute. Modern .38 S&W is loaded down to suit the old back powder guns but the pressure still peaks faster than blackpowder, so you will see more rapid wear with modern ammo.
If you are looking for ammo Old Western Scrounger may stock it.
In various versions the Safety Hammerless was produced from 1887 to 1940.
Although well made and interesting guns, they are not particularly rare, so values are not high. In 2006 the Standard Catalog of S&W gave the following values:
As New $750, Excellent Plus $600, Excellent $475, Fine $300,Very Good $235, Good $195, Fair $135, Poor $75.