Missouri has issued resident CCW's since 2004 (law passed in '03).
Missouri honors all state's permits regardless of residency.
You can carry a firearm in to a restaraunt that sells alcohol, but not a bar that sells food. (51% of the establishment has to be devoted to food sales, making it primarily a restaraunt.)
You may not carry into schools, airports, or venues seating more than 5000 people, or any other place that is restricted by federal law. (Post offices, etc.)
Businesses can post GB ("gun buster") signage, but it must be 1" lettering on a 11" x 14" sign....(even then it carries no legal standing) If you are caught in a posted establishment, all they can do is ask you to leave. You can only be charged with trespassing if you refuse.
Missouri has a "peaceable journey" law which allows ANY citizen (who is lawfully allowed to posess a firearm) to carry a firearm in their vehicle in ANY manner (loaded or unloaded, openly or concealed) without a permit. (Be careful if you choose to carry concealed in your vehicle without a permit....you will be in violation of the law if you exit your vehicle in that manner.)
There is no Missouri state law against open carry. It is legal in Missouri... HOWEVER...Missouri has no statewide premption on open carry. Only concealed carry is fully preempted!
This means any municipality may prohibit open carry. Most don't. Some do. You must fully research each and every municipal code for each and every place you go should you choose to carry openly. Otherwise, you may be perfectly legal in some areas until crossing the street, where you will be arrested.