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The first and most important thing to do is get your driver's license and ensure that it is valid for the state you want to work in. For some roles, especially those driving cars or smaller trucks, this can be a standard license, but licenses issued in certain states might have restrictions which will limit the scope of the work you can do.
If you want to drive larger vehicles, you will need to complete the training and examination to earn a commercial driver's license. The requirements for these can vary from state to state, so check what you need for the state you want to work in and ensure that you maintain a clean driving record.
Most recruiters will expect applicants to have a high school diploma or a GED and any experience with driving will be looked on favorably.
Plenty of delivery drivers choose to expand their skills by training to drive different/heavier vehicles and broadening the scope of the services they can offer. This might include dangerous or specialty loads which can be in high demand.