Anything you want. The 686 can certainly handle any .38 Special loading for the rest of your life and the lives of your children, grandchildren, and great-grandchildren.
Now, are you talking real target use like in a competition (I doubt it - you wouldn't be asking this question if you were a competitor) or just going shooting at the range?
For actual target competition shooting, the most accurate use at short (under 50 yards) ranges, full wadcutters are probably the best but the loaded wadcutter rounds are hard to find and expensive nowadays. Lead semi-wadcutters are very accurate and cut nice holes in the target. You may have difficulty finding these loaded rounds commercially for sale at a good price. Most people who are serious about competing load their own wadcutter and semi-wadcutter rounds.
For just going to the range, though? Most of the inexpensive "blasting" ammo for sale these days seems to be in a round nose configuration. This would be perfectly adequate for going to the range to familiarize yourself with a gun or to just do some routine practice.
A lot of companies make this sort of ammo. You might just ask the sales clerk what he has a good deal on that he recommends for the use you intend. Winchester has a relatively inexpensive 130-grain .38 Special loading with a jacketed round nose bullet that it sells under its USA brand - this is an easy shooting and easy-to-clean-up round. Perhaps you might look for it - Wal-Mart sells it, though it's not always easy to find these days.
Your 686 is a nice gun; enjoy it.
