Getting to the shopping center is quite easy for us. It takes about 20 minutes across the Tappan Zee Bridge which takes us directly to Exit 12 where we can spot The Palisades Center from a distance. There is plenty of indoor and outdoor parking, follow the signs to find a lot closets to the shops you are visiting.
To the best of my knowledge there really isn't much offered by way of public transportation to the Palisades Center. We always drive here. For us its pretty quick and it gives the boys some quiet time, at least one of them sleeps in each direction. If you are coming from the north you are gonna have to hope on the NYS thruway and pay a toll or you can take one of the many side roads to the area, but beware that will take much longer.
In order to get to Rockland ounty from where we live we must drive over the TAppan Zee Bridge. It's an easy drive since there is toll for only one direction. The cash toll is $4.50 for passenger cars. We have E-z Pass so it is easy to zip right through the toll at a nice speed.