Which are the "best?" Specifically those that would allow one to become a part of a greater community.
Moderators: please note that academic journals and more common variety periodicals are not the same thing. As far as I can tell there is no discussion of academic journals so far on Stack Overflow.
~ William Riley-Land
ACM Queue, another offering from ACM has content that's more practical than academic (unlike Communications of the ACM). Dr. Dobb's is another such magazine.
I hate to say it, but journals just don't cut it for me anymore. I used to read several, but have found that an RSS aggregator with some good blogs plus twitter has been MUCH more useful to me than journals. Often, the same guys write in journals AND blogs, and the blog writing is much more prolific.
If you can justify the cost yourself or have an affiliation with some institution to pay for it, an ACM membership with the digital library subscription gives you a print copy of CACM, plus online access to more journals, conferences, and other pubs than you will ever find time to read.
Springer-Verlag also has a web subscription (even more pricey though) to its Lecture Notes in Computer Science series, which is somewhat less cutting-edge than many journals, but tends to have many good introduction/overview/survey type materials which are great for anyone who wants to keep up on areas without necessarily following all the current research.
You should definitely check out any local libraries you have access to...most publicly funded university libraries provide some level of service to the general public and may have current subscriptions as well as extensive collections of past issues of journals. This is way more economical than paying for lots of reprints of anything older you want.