I would like to know if there is a product that will allow me to
answer my business phone with a personalized recorded greeting, which
will play automatically when I pick up the call, and then allow me to
speak directly with the caller.

What I'm looking for is a way to record my own voice saying something
like, "Thank you for calling Morton's. How may I direct your call?" so
that I don't have to repeat this phrase a hundred or more times per
day.  I don't want an answering machine type of thing. Nor do I want a
PBX type of system that transfers callers automatically.  I want to
talk to the callers myself, I just don't want to repeat, a hundred
times per day, the same phrase I use to answer the phone.

I'd like to be able to pick up the ringing phone, have my pre-recorded
message play (and even hear it myself), and then be able to speak
directly to the caller without the caller realizing that the greeting
was recorded and not actually me speaking.

I would like to be able to do this with 3 phones, each with multiple lines.

Just get an automated system that will direct the call on its own
after the caller hears the menu and pushes the button to get the
department they want.  Hey - we can eliminate your job!

Would http://www.grandcentral.com do what you want?