The 357 magnum S&W 686 is a beautiful handgun. My wife and I both use 357's. If you own a 357 then shoot 357. (I have never in my entire life fired more than a couple of hundred rounds of 38 special out of any of my 357 magnum revolvers. This is because the 38 is 1/10" shorter than the 357; and, I've always been wary of, 'fire etching' the chambers.)
Following up lead bullets with a cylinder, or two, of fully jacketed rounds is a good idea; however, FMJ will NOT remove the lead buildup from the barrel or cylinder - Instead it merely irons it flat and degrades accuracy.
If I were to train students to shoot with a 357 magnum pistol by requiring them to show up at the range with $80.00-$100.00 of high quality 357 (or 38 special) self-defense ammo, believe me, I'd be left standing there on the line, all by myself, with nobody for company!
You should train with whatever cheap crappy range ammo you can afford to shoot a lot of - Period! Winchester, White Box; Remington, UMC; Mag-Tech; or Blazer, Aluminum are all good choices. Personally, I don't use lead bullets; and I don't encourage anyone else to, either. This is because I'm always around firing lines; and, over the course of my life, I've been exposed to far too much of this stuff.
Rather than lead, I recommend fully plated bullets like those available from Rainier Ballistics, or Berry Manufacturing. They can act like lead inside the barrel; but, generally, you don't get to breath in the same crap every time you fire a shot.
Once I've got a student showing good control over the pistol I'll switch him to a couple of cylinders of full-blown combat ammo at the end of the session. This technique has worked well for me: Other than occasional comments about, 'more noise' everyone always continues to keep their groups nice and tight.
My number one cheap, 'el crapo', training ammo? Blazer Aluminum
EDITED: Just noticed your comment about gun cleaning! First, you should always wear Nitrile or latex gloves while gun cleaning - Always!
Get yourself: (1) A can of, 'Kano Kroil' penetrating oil, (2) A, 'Kleen Bore, Lead Away' cloth, (3) A Hoppe's phosphor-bronze brush, (4) a large bag of cotton patches, and (5) an old toothbrush - for around the forcing cone.
(A tube of, 'Iosso Bore Cleaner' and a bag of cotton cleaning patches is the equal of a, 'Lead Away' cloth.)
Soak the pistol down with, 'Kano Kroil'. Let it sit for 1/2 hour. (Keep the, 'Kano Kroil' away from any wood grips! Even go so far as to remove them if you have to.) Then begin scrubbing out the cylinder and bore with that phosphor-bronze brush.
Dry the bore and begin using small cut squares of the, 'Lead Away' cloth that you have wrapped around the brush's head. Don't change the cloth as soon as it turns black; it will continue to cut the lead and remove it.
Continue doing this until the patches start to come out clean and you don't see any, 'streaking' inside the barrel when you hold it up to strong light. Do the same thing on the cylinder face. (Don't use, 'Lead Away' on a blued gun - OK! I mean, you can; but, then, your cylinder faces will look like some of mine - bright metal!)
I finish my bores with a liberal coating of Sentry Solutions, 'Smooth Kote' - Yes, it's actually worth the money!