If I am still a dependent living with family in the west coast, but I am temporarily moving and working in another state (FL), what should I prepare and be expecting this year for payments?
Since you are a California resident, you should expect to pay California tax on all of your income, including the income you earn for working in Florida, the same as if you had earned all of the income in California. Florida has no state income tax on individuals, so there is no issue of double taxation.
Your Florida employer probably did not withhold California tax. If not, and if you did not make estimated tax payments to California, you will have to pay all of the tax for the year when you file your tax return, plus penalties and interest for not having paid during the year.
On your federal tax return it doesn't matter what state you lived in or what state you worked in.
Thank you so much! Can I do all of this through TurboTax? Will it be clear how to do so once I input my employment information? Or do I need to be prepared to make payments through other means?
You cannot make estimated tax payments through TurboTax. Go to Estimated tax payments on the FTB web site for information about making the payments to California.