Correct circuit breaker for 12 amp wiring

We recently had a new storage shed built and the wiring is 12 amp. We will only be using tools, etc., that use 110V...NO 220. We have a GFCI just inside the shed, where the power wiring comes into the shed, and the wiring to the power pole panel goes underground (about 30 ft) through the conduit and inside the power panel and the circuit breaker, which is presently a 20A GE breaker. The problem is that the circuit breaker keep tripping intermittently. I'm wondering if I should/could use a 30A breaker instead, since there is a GFCI before it goes into the shed wiring?

 

Okay, by code, this is so wrong and dangerous. That said, let's move on.

A 20 amp circuit requires a 12 AWG wire. 14 AWG can only be used on a 15 Amp circuit.

Underground wire whether in conduit or not is a wet location which means you would need UG type cable or use THHN/THHWN or THHN/2 12 AWG wires. These are listed for use in a wet location.

Next, shed grounding. The shed needs to have it's own grounding system, ground rod connected to the panelboard. the Neutral from your GFCI connection cannot be joined to the ground of the shed.

Now for your problem, A GFCI is there to protect you from injury caused by a short. I (roughly speaking) looks for a 0.6 amps difference between Neutral and Supply (usually black). If it sees that difference, it shuts the circuit down from receptacle and all down stream devices attached to the load side of the receptacle.

In your case, you are using a circuit breaker upstream of the GFCI. The GFCI will not react to an over amperage situation. That is what the breaker is for. This can come from anywhere on the circuit and is usually caused by motors starting up.

Putting in a 30 breaker to protect a 20 amp circuit may get your shed to burn down for you, don't do it.