Converting from NEMA 14-50 or 10-30 outlet to a NEMA 6-15 outlet?
You will need 3 wires.  


A ground wire that is bare, green, or yellow/green striped.
Two other wires that you can distinguish from all other wires. You do not need to distinguish them from each other. They can't be any of the ground colors.  
any other wires, cap them off.  
if any of the two wires are white or gray, you must mark both ends with black tape.  


What I mean by that is if your colors are black-black-white-bare, you must use the black-black as your conductors, bare as ground, and cap the white.  You don't need to care which black is which on a NEMA 6 or 14.  

Because you are using a 15A receptacle, you must get a 15 or 20 amp 2-pole breaker for your panel and fit it. Your two wires go on that. Ground to the ground bus, of course. 

Then fit a NEMA 6-15 receptacle. You can use as many 6-15 receptacles as you want on the circuit, and they often come in a duplex.  Also get the correct cover plate for your box.  (If your breaker is 20A, you must use a minimum of two 6-15 receptacles, so get that duplex!) 



Good chance the wires won't fit on the terminals.  Get 8 inch lengths of copper 14AWG wires (12 AWG wires if a 20A breaker) and pigtail.  You can use wire nuts or Alumiconns. If any wires are aluminum, definitely use Alumiconns. Aluminum wire is fine for large wires 4AWG and up, just terminate them properly.  

Due to the 15/20 exception,  you can use a 20A breaker instead.  However: a) all your wires must be at least 12 AWG, including your pigtails.  And b) you must either use NEMA 6-20 receptacle(s), or more than one 6-15, e.g. that duplex.  6-15 plugs will work in 6-20 receptacles. 

What to do with that NEMA 10-30

NEMA 10 is bad news - it is the "death socket".  It harkens back to the age before grounds, and provides two hots and a neutral.  Sadly NFPA compromised with appliance manufacturers to allow them to ground the appliance to the neutral.  Any trouble with your neutral wire will "float" the neutral at near 120V, and that electrifies the chassis.  You should smash the 10-30 to smithereens and burn the pieces... and convert it to NEMA 14-30 (or anything else with a ground) ASAP.  

But good news for you.  First, you are allowed to retrofit grounds.  You don't have to go all the way back to the panel, just to any nearby ground which is large enough (and comes out of the same panel).  And look! Right next to the 10-30 is a 14-50 with a ground!  That was easy.  

Sometimes NEMA 10 circuits use cables with neutral carried as a bare wire wrapped around the two main conductors like a net. You're grandfathered! and can still use that as a neutral, make sure to wrap it with insulation so it can't contact the ground wire or box.   Or you can sacrifice your grandfathering and use it as a ground for a NEMA 6.  Your call.