Unless you are fired with cause, i.e., because of something wrong you did, you can apply and qualify for the unemployment. Alas, with the sad state of CA state government workers, probably the unemployment payments will not kick in until you see your first paycheck. The office which handles unemployment cases is awfully slow, then they need to conduct an interview with you and with your past employer. If they don't respond, they have to wait so many days etc. You see where I am going.
But by all means apply for it. It will be a bonus when you get it. Don't forget to report every week that you are looking for work and at least send resumes out and record the company names. You need to send them out to the office or enter them somehow to the system using online or phone system. Also, don't forget to report on the first day of your work to stop unemployment payments. Not doing so, is considered fraud and you don't want to find yourself there.
Reach out to your landlord and other people you need to pay before your first paycheck arrives and explain them your situation and tell them that you will pay every red cent you owe them once you get paid and pay the administered late fees and interest if that is the case. Most people are more understanding when you give them a heads up than not finding the payment in their mailboxes on its due date. And take this as a learning opportunity to accumulate a rainy day fund to tide you over at least 2-3 months. You were lucky to get a job immediately. Next time you might not be that lucky. Take it from someone who got laid off under the same circumstances as you, 10 days before 9/11 attacks happened. 6 months of unemployment is nothing to balk at.