Regular Outlet to GFCI Outlet with four wire connections and no ground
Regular receptacles have screws for splits (one switched, one always on or two switched but different circuits or two on separate circuits for higher total power) &amp; for convenience (the way you and most people use them). GFCIs can't do splits and use them for convenience to connect another receptacle only by properly identifying and using LINE vs. LOAD.
Two ways to do this:
Line Only
This is by far the simplest. Pigtail the two blacks together with a short piece of black wire and connect it to the hot screw on the LINE side of the GFCI. Pigtail the two whites together and connect to the neutral screw on the LINE side of the GFCI. Nothing goes on the LOAD side. The new receptacle is protected. No other receptacles on the circuit are protected by this GFCI.
Line and Load
You need to identify which hot &amp; neutral goes back to the panel and which goes on to another receptacle. There are a number of ways to do that - simplest typically is to disconnect all 4 wires, turn the breaker on and see which wires cause a non-contact tester to light up. Connect the &quot;panel&quot; (may be other receptacles in between, and that's OK) wires to the LINE side and the other wires to the LOAD side. Now the new receptacle and everything following it via the LOAD side is protected.
Note that the second method is generally OK and provides additional protection. However, if any of the LOAD side receptacles are things you don't want on a GFCI - e.g., anything like a refrigerator that would have problems if there were a trip of the GFCI and you didn't know about it for a while - then go with the first method.