NJ
I will echo what 1fly mentioned. Depends on where you are. I have lived in South Jersey for the past 35 years so quite familiar with that part (as a gun owner, I am glad to live in PA now). NJ gun owners have lots of hurdles to overcome when buying a gun, so if your not familiar with the states gun laws, start reading up and asking questions. You'll need a NJ Firearms Purchaser ID card and a Pistol Permit for a handgun and either can take 3 months to process. Welcome to NJ.
South: Bob's Little Sport Shop: Owned by Bob Viden who is also on the NRA board of Directors as I recall, although may serve the NRA in another capacity right now. They have a shooting range in-house. I have purchased many guns through Bob and his family. I support Bob's shop as much as possible because he does take an active role in defending the rights of gun owners and especially gun owners in NJ in one of the most restrictive states in the nation.
Central: Bordentown Area: Harry's Army Navy or Sportsmans Center.
I do not frequent North Jersey, so can't offer much help there. There was a place called Ray's Sport Shop, but I believe it may no longer be open.
On a common gun like a M&P, you shouldn't have much of a problem getting one wherever you go. It is a good idea to develop a report with a local shop though. It can help you out in many ways.
If you are looking for a place to shoot in South Jersey, it doesn't get much better than South Jersey Shooters in Winslow (outdoor pistol/rifle) or if you like trap/rifle/pistol, Square Circle Sportsmen in Berlin.
IC