Greetings,
MY wife and I we are both new to the target practices at the gun range and to the selection of ammo for the defense purposes.
We would like to practice more often however, we are not sure what type of ammo ( including the grain weight ) would be most economical for both guns in target practice sessions and for the possible defense events.
Also, what do we do with the accumulating quantity of empty shelves?
We appreciate Forum's comments.
What do "shelves" have to do with ammunition?

Sorry: you probably meant to type "shells". You really mean empty cartridge cases, called "casings" on TV cop shows that don't know much about guns. The word "shells" normally means shotgun ammunition.
Because it is obvious in your post that you know little of firearms, you will not be reloading the cases. If you know friends who handload, give them the empty cases. Or, most shooting ranges have barrels in which to toss the empty cases, sometimes called,"brass", as they are normally made of brass, although many today are nickel plated.
As for ammo, your M-686 will digest any commercially loaded .38 Special or .357 Magnum cartridges. But you will not want to fire .357 ammo for home defense, as the muzzle blast is so loud that it may damage your hearing if fired indoors.
The best .38 ammo for both guns for defense is the former FBI lead Plus P load cited by another member or the Speer Gold Dot cited by another. I load 125 grain Gold Dot in my longer-ballelled guns and the special Short Barrel 135 grainers in snub and three-inch barrels. But Federal's Hydra-Shok 129 grain also has a good reputation and may be easier to locate.
If you use the .357 outdoors, perhaps on a camping trip, the Winchester Silvertip 145 grain .357 or the 158 grain Federal Hydra-Shok are my preferences.
You will get as many opinions as those who post. You asked a question akin to, "What is the best car?" But the opinions that I've formed are based on many years of researching the matter and I have talked with engineers at several ammunition companies, which was possible because I was a gun writer. And another member who was a police detective near me (I met him once) recently posted here that Dallas PD had EXCELLENT results from both the lead .38 HP ammo and (especially) the 145 grain Silvertip .357, which was also used by the FBI when they still allowed .357 revolvers.
A spokesman for the Federal ammuniion company also told me that their 158 grain Hydra-Shok .357 performs very well on deer within reasonable ranges. It's very accurate in my guns.
However, the 158 grain .357 offerings from Winchester and Remington and Federal's "normal" JHP bullet in that caliber are good, too. The 125 grain bullets are very effective on humans, coyotes, etc. One man I know about used them on alligators, too, with satisfaction. But they erode the forcing cone and topstrap more and in medium-frame S&W .357's, they may cause cracking of the barrel, usually on the thinner flat portion at the bottom of the barrel, at the forcing cone. But your larger M-686 has no thinner part of the barrel there. It was designed especially to be longer lasting with all .357 ammo.
For .38 practice ammo, I usually buy either Remington or Winchester 150-158 grain lead bulleted loads, whichever is cheapest where I find them. I avoid ammo from Eastern Europe and from China, as I do not know how good it is or to what standards it may have been made.
If your dealers carry Black Hills brand ammo, that is very good, too.