I've been pretty happy with the Enhanced Payroll that comes with QuickBooks Online. It is inexpensive and all your payroll details are right there in QuickBooks. It's good enough for companies with just a few employees. QuickBooks Online Payroll can e-file and pay, but you must push a button for this to happen. I prefer this to automating everything, because it forces you to make sure your payroll is complete, and to make sure you have cash in the bank! The individual paychecks will show up in your check register, rather than one big withdrawal.
I've had many clients satisfied with Paychex, Gusto, ADP, and Hill Country Payroll. They offer better HR support and help with complicated situations. If you use one of these services, The payroll report needs to be entered into QuickBooks Online as a journal entry. I can do this as part of my service.