In short the answer I've reached is get your CCW permit even if you only plan to carry in your car. Otherwise you will most likely fall foul of the school zone law.
You can carry in your car and conceal the weapon. A rifle or shotgun must have an empty chamber, mine is always unloaded, not sure if you can have any in the magazine. Handguns can be loaded and chambered.
One thing I found out recently is the restriction of the 'School Zone' to carrying in your car.
Maybe out of date but here is some info
It is illegal to bring a gun with 1,000 feet of a school, with certain exceptions.
Gun Free School Zones Act. 18 U.S.C. § 922(q):
(2)(a) It shall be unlawful for any individual knowingly to possess a firearm at a place that the individual knows, or has reasonable cause to believe, is a school zone.
Under 18 U.S. Code § 921(a)(25) The term "school zone" means:
(A) in, or on the grounds of, a public, parochial or private school; or
(B) within a distance of 1,000 feet from the grounds of a public, parochial or private school.
(26) The term "school" means a school which provides elementary or secondary education, as determined under State law."
However the certain exceptions mentioned above include CCW permits issues with background checks. More from http://gunowners.org/fs9611.htm
THE "GUN OWNER REGISTRATION EXEMPTION:" The "gun free zones" law exempts CCW (Carry Concealed Weapon) holders who live in a state that requires a background check before the issuing of a permit.
So if you only plan to carry in your car, it's a very good idea to still get a CCW permit as well! 5 years in prison just for driving within 1000 feet of a elementary or secondary school. Good luck avoiding all them, especially as they include home schools!
Another thing that make this School Zone law impractical is, 'weapon' is not just defined as guns but knives too. If you don't have a CCW and drive to your next hunt (even if it's a bow hunt) you'll have to plan the drive in and out of each city really, really well to not fall foul of the law!
Short answer again, get a CCW permit.
Not doing 5 years for driving with your gun...Priceless.
(not legal advice, and may be inaccurate!!)