Coin World and Numismatic News For us Americans that is. Some people prefer one over the other, I subscribe to both.
I also get Coin World and The Numismatist. Both are good. The Numismatist is the official publication of the ANA and comes out every month; Coin World is published weekly. Walt Southward
For newcomers, the best bets are the newsstand coin magazines rather than the coin newspapers, which are more for industry insiders, dealers, and advanced collectors who want to keep up with the latest news. The magazines have more background information about coins to deepen your appreciation for your collection and to give you ideas about new collecting areas you might want to explore. The two big newsstand magazines are Coins and COINage. Lots of meaty feature articles, columns, and so on, all targeted toward collectors. Browse or buy a few issues then decide which to subscribe to. Another good magazine is the Numismatist, which you receive for free by joining the American Numismatic Association (ANA). The articles in it used to be more scholarly, though the magazine had recently undergone a "populist" transition. The best magazine for ancient coins is the Celator. The coin newspapers, Coin World and Numismatic News, can still be excellent reads, with some background information about coins along with price guides and so on. But they're more for dealers and advanced collectors, with lots of information about goings-on in the coin industry.
I'm a charter subscriber to World Coin News now in its 30th year. A lot of info for world coin collectors! http://www.krause.com/static/coins.htm
As a curiosity and because it's the official magazine of Error World Coin Groups,might I add Mint Error News ? It's available online at http://minterrornews.com in PDF files,but this is also available as a printed magazine absolutely free by emailing editor@minterrornews.com Ya gotta love it! 100% free like everything in EW Groups. ~ Jim