A good place to start when you are designing a circuit that you are unfamiliar with is Wikipedia. They explain that the Wien Bridge Oscillator needs a non-linear element to force the feedback ratio to be exactly 2:1. An incandescent lamp is commonly used to accomplish this.
I found a very old chassis mount incandescent lamp that is probably about 5W @ 120 VAC. I measured the resistance at various low voltages (since I don't have a milliamp meter that will go below 1 mA, I put a resistor is series and measured the voltage across the resistor). The bulb needs to have a reasonable resistance (~300 - ~3k) at 1V.
One volt (DC or RMS) or slightly less is a good target voltage for the bulb. The output Vrms will be 3 times this and the peak will be 1.414 times that, so 1 volt RMS at the bulb will be 4.2V peak at the output. If your OpAmp output won't go rail-to-rail, your target bulb voltage may need to be a little less than 1V (or use higher than +/-6 OpAmp voltage). Now, pick your feedback resistor to be twice the resistance of the bulb at the target bulb voltage. For my bulb, the resistance at 1V is a nice even number (500 ohms), so the feedback resistor is 1k.
Good decoupling is important. I have a 100 uF and 0.1 uF ceramic on each rail to ground.
Next, I found a new issue. If the loop recovery time is near the thermal time constant of the bulb, you will create an amplitude modulator. I made a nice 6 Hz amplitude modulated sine at one point. If the oscillator frequency is higher or the OpAmp has a higher bandwidth it doesn't modulate. If you are making a product to sell, it will be difficult to prove that AM oscillations won't happen under all circumstances.
This is a finicky circuit, if you want to use this in a product, you should reconsider. Hewlett and Packard created the famous tube Wien bridge audio oscillator in the 1950's, but they were damn smart, and there are better ways to do this now.
simulate this circuit – Schematic created using CircuitLab