We did this on our own, and the set up is a pain, but after that it's very, very simple. I agree, no need to pay big bucks to someone else if you don't want to.
First, you will need to spend a lot of time educating yourself on the relevant federal laws (IRS publication 926) as well as your state laws and unemployment laws. You will have to apply for both a federal and state employer ID and figure our how to set up a unemployment account.
This is all one time or year stuff, however, and once you get through it, it's simple.
If you are not withholding income tax (not required) you don't even need a computer program. Just deduct the current employee fica taxes from each paycheck (currently 6.2% for soc security and 1.45% for medicare) Give your nanny a pay stub each week showing hours work, gross pay, deductions and net pay. Be sure you pay to the IRS, either through increasing your withholdings or quarterly payments, double what you withheld from the check (to cover your portion as well)
Turbo tax will walk you through the forms at the end of the year to file this properly. It will become part of your tax bill. You set up a separate account in January with the soc security administration to report the withholdings for the employee which will allow you to print out the W-2.
If you agree to withhold income tax, the calculations are a little trickier. We used the program NannyPay. It was about $100/yr and also walked you through these steps, calculated pay each week, and allowed you to print a pay stub. We used it for about 3 years before realizing we didn't need it anymore since our housekeeper didn;t want income tax withheld.
Good luck!