556 is pretty expensive atm. The S&W M&P is more accurate because it is specifically made for 22lr. If you use use a 22lr upper with your 556 barrel there could be some looseness in regards to the bullet+ barrel size which can effect accuracy. (Not much.) But it's not tolerable for target shooting.
It's your call really, AR's and their variants are overpriced now. $600 for an M&P isn't too bad but I'd still recommend the 10/22 as a starter rifle. the price difference can include a decent optic and a few bricks of ammo.
Factory 10/22 $200 (Take down 10/22 is $330 or so.)
Bushnell TRS-25 Red Dot $80~$100
Nodak Spud Tech Sights $100
Some Reliable Magazines (Standard 25 rounders are tough to get atm.) $90 can get you 3 at normal price. 10 round magazines only cost $12.
Auto Bolt Release $15
You may also want to check this out when you get good enough at marksmanship.
http://www.appleseedinfo.org/
I plan on it this spring. A lot of people use 10/22 for it.
Google Liberty Training Rifle for the 10/22.