There are many deadlines that you’ll need to adhere to, so I strongly recommend that you:
Use a payroll processor: like Gusto or Justworks. I love Intuit QuickBooks Online but stay away from Intuit Online Payroll. Using a payroll processor will help you meet all of your payroll tax requirements, and greatly reduce the risk of getting these payroll tax forms wrong. The government will penalize you heavily if you don’t make payments on time.
Scan for state and local requirements: As for all of the other state and local tax requirements, check out these two lists:
Work with your CPA: those lists aren’t exhaustive, and you may have a unique situation that wasn’t covered. Maybe your specific business requires additional permits. Your CPA can also help you determine your tax “nexus,” which is a fancy way of saying where you need to file tax returns because of your Sales, Property/Rent, and Payroll.
Hope this helps!