Basically any 38Spl ammo you can scrounge up cheap
will work great in that thing.
What I personally would do is first, figure out which combat loads you're going to shoot in it, and then find a practice load that is ballpark similar in terms of weight. Even when the power level is way different, loads of the same weight will tend to print to about the same vertical elevation. Not the terms "about" and "tend", as there are exceptions.
Still, 38Spl practice ammo of 130grains is easy to find and will be close to the same elevation as 357Mag 125grain combat loads with no adjustment of elevation needed. If you're going to load 158grain JHPs for home or street defense, then 148s and esp. 158 38Spl practice fodder will go to around the same place.
Follow?
So let's talk about combat ammo, even though that's not what you asked about it's a related subject.
In a large 4" barrel 357, I think there's a lot of merit to loads based on the Speer Gold Dot 158grain 357 load. These are typically rated at around 1,250fps or similar - you do NOT want the "full house 158 hunting loads" at 1,500ish. This fairly mild 158 expands well at speeds between 1,100 and around 1,400, with the "sweet spot" right around 1,200 - 1,300, which is generally what you'll get with this load and a 4" barrel. These are loaded by Speer, Proload, Black Hills and Georgia Arms, the latter is probably going to be the cheapest esp. if you buy by their 100-count. You then score some 38Spl lead 158s for practice. The 158 Gold Dot will feel more controllable than the hottest 125grain full-house 357s like the Cor-Bon, Remington and similar. 158grain practice fodder is easier to find than 125s, although 130 38Spl is easy to find and usually works as a substitute.
If those 158 Gold Dots prove to be too much of a handful (how strong are your wrists, how much experience do you have?) then the next step down in power will be the milder 125s...Proload has a 125grain "Tactical Lite" 357 meant to be controllable in small snubby 357s, and will be very easy to cope with in a 686. Remington has something similar, the Golden Saber.
Next down from those are the various 38+Ps.