What part of my time is my employer legally required to pay me for?

Are you an employee or an independent contractor? If paid by a W-2, you are certainly an employee; if a 1099 is used, you may still be an employee, because even if someone is called an "independent contractor" and paid on a 1099 basis, if they meet the criteria to be considered an employee, the law requires them to be treated as such no matter what their employer (or they) want: you can't turn someone who *should* be an employee into an independent contractor unless you actually structure how they work, etc. so they are independent.
So as stated, if you will get a W-2, you are an employee. If paid on 1099, you will still be an employee if your employer sets your hours and can tell you when to work; directs you (or at least has the power/right to direct you) in how to do the tasks; and generally manages you the way employees are managed. For the long description of how to tell an employee from an independent contractor, you can find information on both the U.S. Dept. of Labor and the IRS websites. For the short description, if you are more like an administrative assistant (who does whatever the boss wants, when the boss wants) than like a tax preparer or CPA (who get the tax information, but then works on the tax return when and how he/she likes, coming back at the end with a finished product and a bill for services) you are an employee.
This is important because:
1) An independent contractor is not protected by labor laws. As the term "contractor" implies, the way they are paid is set by "contract" or agreement, even if an oral (unwritten) one. They are paid whatever they agreed to be work for, and if the employer offered to pay you only a set or maximum amount of time per task no matter how much research or preparation is needed, or how long the task takes, and with no compensation for administrative, overhead, etc. time, and you agreed to work for that, that's all you are entitled to. Negotiate a better deal going forward or find a better employer.
2) But if you are an hourly (not salaried) employee (i.e. an employee paid by time), you must be paid for ALL time worked--including research, preparation, status reports, correspondence, timesheets, billing, etc. And when you work more than 40 hours ina week, you'd have to be paid overtime.
If you believe you are an employee, you may be owed a considerable amount of back pay, even overtime as applicable. You may wish to contact the state department of labor to see if they can help you collect, or consult with an attorney about possibly suing.