The wires connected to the sensor are special wire, but the wire from the connector all the way to the ecu is just standard copper wire. You will not be able to solder directly to the O2 wires, they tend to not conduct heat... I would recommend setting it up with a factory connector anyway for ease of changing.
On that 3 wire sensor, the white wire is the signal wire and the 2 black wires are the heater wires. connect one heater wire to a good ground point and the other to the heater control power, normally an ignition controlled fused 12V source, it doesn't matter which black is power and ground.
The reference ground for the signal runs through the sensor housing into the exhaust to the engine. On a 4 wire sensor it would have an extra ground wire for this that would be connected to a separate clean ground point (not the same wire for the heater ground) but that doesn't matter on your 3 wire sensor...