So What should you do?
1) If you can get the other parent to agree to your moving, you should get it in writing. I would suggest doing a stipulated order with the court so that everything is official. You do not want them claiming later that you said the move was temporary, or that they never said it was okay, etc. Too much hassle on your end. Make sure you have a written stipulation, signed by both of you in front of a notary (so they cannot claim later it was not them who signed it) to submit to the court. This will allow you to feel confident in moving before you actually have the order signed by the court, since it can often take time to get the order back from the court signed off on. Be specific in the written agreement regarding your rights to relocate - for how long? (temporary or permanent), do you now have sole right to determine residency within the United States and you can move again in a year to a different state if you want, etc.
2) If you cannot agree then you need to do a Request for Order asking the court to allow you to move. In Alameda County expect the judge to send you to mediation on the day of the hearing. Be as cooperative as you can with the mediator. Almost 100% of the time, the judge will listen to the mediator's recommendation. If you cannot agree at that point and the mediator does not think you should move, expect not to be able to move for a couple of months so that you can do additional mediation and have a final hearing afterwards if you cannot agree.
A lot of the mediator's opinion will depend on why you are moving - if the reason is just cost of living or you have family there then you will likely have to wait. If you have had sole physical custody and have a job opportunity that you could lose in the other state, the court will be more likely to rule for you to be allowed to move sooner.