Payroll is sufficiently complex to justify outsourcing it to someone else. You don't want to make a costly mistake on taxes, etc. We've outsourced payroll almost since day one. There are a number of companies that will handle payroll, even for very, very small companies.
Yes, outsource it. I've worked in payroll before and it is unforgiving.
Tell them to talk to the bank that they are going to use for the business. The banks usually have a connection to someone local that does small business payroll. No sense paying for ADP until you need all they offer.