Getting married this year. Between us, we have 2 mortgages and 4 children. Should we file separately or together to reap the greatest tax advantages?

It is almost always better to file jointly. Several credits/deductions (like the dependent care credit for example) are not allowed when you file separately.
An exception would be if you live in Ohio, then filing separately might be better.
If you are using tax software like turbotax (the desktop version), you can do a "what if" scenario and see for yourself, which filing status is better for you.

While I agree that filing jointly is probably better, you will not know for sure until you model the return BOTH ways and compare the results. This can be easily done using tax software, or any competent tax professional will do this modeling when he/she prepares your return.