Those top-break guns look kind of old and antiquated, but S&W carried them in the catalog up until just before WWII. I think a lot of them still serve as nightstand and desk drawer guns for people who inherited them from parents and grandparents.
There is a trick to shooting them well. It obviously takes a long trigger pull to make one fire, and since the pull is a little stiff there can be some accuracy-destroying unwanted hand and arm motion. It takes a lot of practice to make the shots land close to where you want them. But take the unloaded gun and pull the trigger slowly, paying attention to what you feel. Just before the hammer is released, you notice a kind of catch or brief moment of resistance. That is there by design. If you train yourself to pull the trigger to that point, then pause and aim before completing the trigger pull, your accuracy will greatly improve. That resistance point actually functions as a kind of make-do single action mode, allowing you to stage the trigger under finger tension before pulling through the last 1/16 of an inch.
Modern .38 S&W ammo (NOT .38 Special, so don't make that mistake) is fine in those guns. I have one that is about as old as yours and shoot it now and again.
You can spruce that gun up a bit with some fine bronze wool and a good gun oil like Breakfree CLP. If you want, you can even use a good nickel polish to brighten the remaining nickel and scour the stain off the exposed steel, but don't overdo it. After you get it cleaned up, just keep the gun oiled to prevent oxidation from starting up again.
Welcome to the forum!