It depends. In all likelihood, you will not need to report the sale of your home to either Federal or state if you had less than a $250,000 gain on the home, and $500,000 if you are married filing a joint return. This is not that you sold the house for 250,000. This means if you made $250,000 on the sale. If this is the case, you will not need to report to either Federal or State (California). Here is an FAQ with more information: https://ttlc.intuit.com/replies/3300213
The purchase of the new home will be entered in with your itemized deductions: mortage interest, points, mortgage insurance premiums, real estate taxes, etc. If the states allow itemizing on a part-year resident return, these deductions will be factored in.
The above information is based on if the two homes were primary residence homes, in other words, where you lived. If they were not, especially the house you sold, the tax treatment is a bit different. The above FAQ will help you to determine this as well.
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