You can connect a solid state relay directly to the Arduino outputs. The input draws only a few mA so no resistor is required. Connect the + input terminal to a digital port pin, the - to ground so that a HIGH on the pin will turn on the relay.
The transistor controlling the fan should have worked, but DC motors, relays, etc. require a reverse diode across the winding to protect the rest of the circuit from high voltage switching transients. Since your circuit does not show one, you may have damaged some of the components.