Do I need to save my receipts if I work out of town? And do I need to save the mileage as well?

If you are a W-2 employee, then job-related expenses are not deductible on your federal tax return, although there are a few states in which you may be able to use them on your state return.  If you are a W-2 employee, you should be saving receipts to provide to your employer for possible reimbursement for the expenses.

If you are an independent contractor, then your expenses go on a Schedule C when you prepare your return and enter your self-employment income.

W-2 employees cannot deduct job-related expenses on a federal return.  Job-related expenses were eliminated as a federal deduction for W-2 employees by the tax laws that changed for 2018 and beyond.  Your state tax laws might be different in AL, AR, CA, HI, MN, NY or PA.

If you live in a state that lets you deduct job-related expenses, the information will flow from your federal return  to the state return, so enter it in Federal>Deductions and Credits>Employment Expenses>Job-Related Expenses

https://ttlc.intuit.com/turbotax-support/en-us/help-article/tax-credits-deductions/employees-deduct-job-related-expenses/L5b07gqy7_US_en_US?uid=lwhvwyxj