To elaborate on this:
American Sign Language, grammatically, is similar to Mandarin. (I know, that surprised me, too.)
It is not Signed English -- basically, just using ASL signs structured like English communication. It's basically the pidgeon version of sign language used to communicate with the hearing, who often have a hard time making the grammatical leap to true ASL. Native ASL speakers are almost all able to understand and communicate with Signed English, since they're familiar with English grammar from reading it constantly.
I started with Signed English and am slowly learning more ASL grammar as time goes on. Most of the time, when I'm communicating with deaf people I know, I do a sloppy mix of both. I think of it as little kid sign.