If you want to catch up with the current continuity, look for books set in the New 52 universe. Most people think of the Justice League when they think DC Comics, so here are some suggestions that will catch you up on the New 52 version of the Justice League:
If you're more interested in the previous continuity that occurred for most of the 2000's, here are some recommendations for that version of the Justice League:
Trade paperbacks are collected editions of popular story arcs, so it's a great way to catch up on major plotlines. The downside is that they don't always include every issue and therefore might have some missing gaps.
If you want character-specific titles or something else entirely, I would recommend going to Amazon (or your preferred online retailer) and search for the name of the character you want and the acronym TPB (ex - "Batman TPB"). On Amazon, you can also include the search term "New 52" to look for items specific to that continuity. You generally want to look for titles like "Batman: Volume 1" when looking for the core stories.
Alternatively, if you just want to catch up on the whole New 52 thing, I also recommend the following animated films:
The first is the adaptation of "FlashPoint", which is the event that changed everything to the New 52 continuity. The second is the adaptation of how the Justice League first formed in the new continuity.