It can be complicated, but it doesn’t have to be. You have two ways to go with this. You can pick on-going series or pick up trade paperbacks.
Let’s look at on-going series first.
You’ve got the right idea in trying to find books that are starting off brand new so that you don’t have to know any backstory to understand what’s going on.
The New Avengers is a good choice. It’s a new start with a new writer–Brian Michael Bendis–who got to hand pick the team members he wanted to write about.
JLA: Classified will be telling previous stories of the JLA. The advantage is that there won’t be any long term storylines to follow. The disadvantage is that it may require some previous knowledge of the JLA to follow along.
Iron Man just started with a brand new #1, and no knowledge of the character is required to follow along, and with a top notch writer (Warren Ellis).
Captain America likewise is getting a new start with a new writer (Ed Brubaker).
Legion of Superheroes is also starting with a brand new #1. Again, no previous knowledge is necessary to jump right in–the Legion universe has been “rebooted”, meaning that they’ve thrown out all previous continuity and started brand new from day one.
For other series, the best thing to do is look for a good “jumping on” point. Usually, this means starting at the beginning of a new story line or when a new creative team takes over.
JLA #107 would be a good example. The new creative team started a new storyline, with no detailed background knowledge necessary to get into the story. JLA is also a good book for the casual reader because of its lineup. Five of the seven core characters (Superman, Batman, Wonder Woman, Flash, Aquaman) have their own books, so the writers can’t really engage in a lot of soap opera type personal drama, and are forced to concentrate on superhero adventure.
Hulk #77 would be a good place to get into that book–the previous writer just wrapped up a very long, complex storyline. The new writer, Peter David, is one of the best writers the Hulk ever had.
You’ll want to stay away from both of the big two’s core frachises. The Batman family of books is a complex soap opera that would be very daunting for a newcomer to delve into. If you are a fan of Batman, however, there is an alternative.
Likewise, I’d advise against trying to get into the core X-Men books from Marvel. It’s also a complex soap opera, but with a good deal less coherency between the various books (there’s roughly a dozen). Each writer seems to be doing whatever he wants with whatever characters he likes, without regard to what the others are doing.
The exception to this is Astonishing X-Men. Sharp writing by Joss Whedon showcases a small, effective superhero team with much less soap opera-ish crap hung onto it. The first six issues are available for a reasonable price (and will soon be released in trade paperback), and a new story arc starts with #7.
Trade Paperbacks
Trade paperbacks collect multiple issue (usually 4-6 issues) storylines. Every storyline from the current run of JLA is now, or soon will be, available. The big advantage of trades is that you get the whole story at once. JLA is a good one for this, especially since, unlike a subscription, it enables you to skip the truly horrid, and hopefully soon to be forgotten, Chuck Austen storyline (Pain of the Gods) that was just completed.
If you like Batman, trades are by far the best way to go. The Batman writers love long, complex crossovers that require you to buy a dozen different titles to get the whole story. The most recent was War Games, which was 24 issues spread out over some 10 different titles. By far the best way to read this is to wait for the trade paperbacks (there will be three of them) and buy those.
This arc wasn’t even the longest. A few years back, there was an arc (No Man’s Land) that covered some 40-50 issues. It’s a good read, and available in trade (you’d want to get Contagion, Cataclysm, and No Man’s Land books 1-5.
Others have covered the Ultimate line of trades (Spider-Man and X-Men), so I’ll just agree with them.