What you want is a switch that looks as follows:
This particular unit comes with pigtail leads that make it easy to connect into the existing wiring in a light fixture.
The switching configuration is specified as:
Single Pole Triple Throw, OFF-ON(P)-ON(N)-ON(N+P)
The switch/outlet combo you have could physically work, but as pointed out by @Speedy Petey, it would not be code complaint since all outlets in bathrooms need to be GFCI. Consider replacing it with a GFCI/switch combo such as this.
The wiring you have should work. One of the cables (which consists of one black and one white wire) is from the mains, and the other cable (black and white) goes to the existing light fixture. You need to determine which is which. The easiest way to do this is with a non-contact tester.
With the power on and the old switch turned off (first make sure no bare wires or contact points are touching anything metal), check to see which black wire causes the tester to beep. If they are already disconnected, just check each wire separately. The one that beeps is the constantly hot wire from the mains. The other black wire is a switched hot going to the fixture. Turn the power off at the circuit breaker and mark the constantly hot wire with a piece of tape. Disconnect the old switch.
Look at the new switch/outlet on the right side (the side with the black screws). If there is a metal connector showing between the two black screws, you only need to attach the hot wire to one of them. If there is no connector, you need to attach the hot to both of them. This can be done by adding two short pieces of black wire (called pigtails) to the hot wire and then connecting the other end of those new wires, separately to each screw.
Now connect the other black wire from the box (the switched hot to the fixture) to the brass screw on the left side of the switch.
If there is no green or bare wire coming from the existing cables, you cannot properly ground this device unless the box is metal and the cables are armored (covered in a spiral metal casing). If there is a green or bare wire, it should be attached to the ground screw. If the box and cables are metal, the metal attachment strap of the device provides a form of ground (but not the best).
If the cable is not armored and there are no ground wires, the outlet is protected by the GFCI circuit, but it is not grounded. It needs to be marked on the outside as No equipment ground.
Images and links are for illustration only, not an endorsement of goods or sources
Best Answer
Exactly the switch you already have.
You look at your wiring -- and in doing so you ignore ground wires (which are green typically). Ground is connected everywhere, but it doesn't participate in the normal wiring, so we don't talk about it. If you see
With 3/4 way switches, which screw is on brass vs. black is very important. The physical position of the screws is useless and misleading (since every switch puts them in different places!!) Also, wire color is useless and misleading - the screws they go to are the key.