How do I operate a petcock on a 1994 Kawasaki Concours to start and run. It has pri, on ,and res

Sounds like a malfunction vacuum petcock. Possibly just a cracked or leaking vacuum line operating the vacuum diaphragm. With the choke on there is enough vacuum to operate the petcock , but as it warms up the reduced flow from the partially closed petcock feeder line causes you to run out of gas when trying to drive at speed. Check the vacuum line for damage and replace if necessary. Check the fuel petcock for internal damage. If vacuum diaphragm is cracked or has a pinhole replace the petcock assembly about $75)(K&L may have a rebuild kit about $30).

You have a vacuum operated petcock....so that when engine starts and draws vacuum...a diaphragm lifts in the petcock to supply fuel to carb. When the engine shuts off...vacuum is lost and the petcock closes.
If you run the tank empty....in order to get fuel into the carb and get the bike started initially....you can open the petcock without engine vacuum by rotating petcock to the PRIME position. Once started return the petcock to the ON position. If you leave it on prime....with engine off...fuel may leak into carb and run into engine and contaminate crankcase oil with fuel.

What you are dealing with is a vacuum petcock with a bad vacuum diaphragm , in this case fuel is bypassing the petcock via the vacuum line and filling the engine (oil) with gasoline.
If you have a standard petcock that is leaking (not shutting off properly) or just left in the on position a stuck float needle will also fill your engine (oil) with gasoline. This can happen with a clean carburetor and a good float needle. It just sticks and you have to change your oil. If you have the kind of fuel petcock that allows you to turn it off when not in use , always turn off your gas when not riding.
You will have to change your oil if it is fouled with gasoline. I suspect a vacuum operated petcock is at fault. With standard petcocks I have seen new motorcycles in perfect order stick a float needle ( Kei Hin Carburetors ) and fill there engines and air boxes with gas/oil mix. On vacuum petcocks the diaphragm that shuts off the fuel flow is usually leaking and at fault.

( PRI ) is the primer position for the petcock ( fuel valve) . This position is used for a couple of reason. this kind of valve works off vacuuum from the carbs, but if you run out of fuel you or remove the tank you can't create vacuum, so putting the valve into the PRI position allows fuel to flow inton the carbs without vacuum so you can start the bike. Once the bike has started you should switch the valve back to the ON position ON= normal position, PRI= primer position, RES= reserve postion. I hope this helps!!!

There is no OFF, you have RES toward front , ON is all the way down, and prime toward back, PRI is used when it has been sitting for awhile. RES and ON wont let gas out unless bike is runnig. There is a vac hose on back of petcock that lets ON and RES work it must be connected..

up (res) is reserve, down(on) is primary, center(marked as pri is for prime, used for if ran out of fuel or drained the carb or been sitting for a long time). There is no off as the fuel system is vacuum fed instead of greavity)

Remove the petcock valve and dissasemble and clean it may have a clogged passage or this type may have bowl with a screen in side that is dirty, clean the valve with gum out and blow out with compressed air if possible.

i'm assuming you are talking about the petcock it should be pointed down to on PRI is prime and should only be used briefly to refill carbs after running out of fuel. res is for when you run low you don't wanna use res all the time cuz when you run out you're out and have no real warning although the kat does have a fuel guage it is still sucking all the crud off the bottom of the tank