If your former employer did not withhold taxes, you will not need to make any adjustments. You may want to check on your paycheck stubs, the amounts would have to reported there as well. When you are hired, you complete a W4 to indicate if you are exempt from withholding.
You will only get in trouble if you report a W2 with information that is not on the W2 you received; since you entered what you have you have done nothing wrong. You can wait and see if a W2C is issued- then you can amend your tax return to make the corrections. Your employer is required to withhold unless you indicate that you were exempt, so if you owed taxes, you ay want to update the information with your employer. (It is always a good idea to review this information periodically and make any adjustments as needed). Some states use the same W4 information, but some have a specific state form. Once again, you should fill one out when you start a new job and review as needed.
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