All I shoot in 38 special / 357 magnum is lead bullets , I even cast my own , since 1967 .
Any factory ammunition will be fine even the ones with lead bullets, don't be afraid to try several weights and makers.
357 magnum loads will usually be jacketed because of the high velocity.
Accurate and pleasant to shoot are the 148 grain target wadcutter ammunition. +P 38 special loads will be between 38 special and 357 magnum. 357 magnum come in all weights, I like 158 grain but I'm old school and 158 grain was the standard weight for years.
Availability has a lot to do with what you buy. Go to store, gun store employee's can offer guidance , pick up 2 to 4 boxes of different ammo and try them and see what you and your gun like.
Good luck , be safe and have fun,
Gary