It is not unusual for government employees or school systems to not withhold Social Security taxes, because they are paying into a state-run retirement system instead.
However, it is unusual not to withhold federal income taxes, because relatively few people who are employed are exempt.
You need to look at your W-4 (the document you filed with your employer when you were hired) and make sure that you did not mark yourself as "exempt". Take a look at IRS Tax Topic 753 for more information.
At the least, you want to visit with your HR department to understand why nothing was withheld in either case, and whether you are eligible for exemption. You may need to file a new W-4.
If you are liable for federal income tax and you had no withholding, then you will have to pay the full amount due when you file. And the IRS will likely penalize you for not paying through the year like most people. This is why it's important you straighten this out as soon as possible.