The best way to learn what has happened since the end of the movies and experience it would be to run Archives available from Org bookshops in Richland. These are critical missions from all orgs from chapter 1 all the way up to the last update and give a good basic overview of what has happened.
There are only a couple of problems with that; each chapter has 3 sub chapters and each subchapter has 5 missions to do with the story. We are just about finished with chapter 10 now so that is over 150 missions per Org (I say over because chapter 1 didn't follow the 5 missions per subchapter rule, it has more) and if you want to get the broad overview you can do all three org's archives (450+ missions). Each mission is set on hard too so all NPC's will be a level higher than you.
Also you can't relive Live events in which the storyline can progress between critical/archive missions but the archive usually explains what has happened.
It's a long grind but there are incentives; Org clothing and 2 min FX rewards in exchange for Archive completion tokens, Good XP better than regular missions and doing an opposing Org's archives will not affect your reputation.
The Live Events happen on a daily basis (weekdays) usually happening evening time MCT (Megacity time, which for me is GMT - 8 so events usually take place around midnight for those of us in the UK), but are spread between the 5 orgs on the three servers. It is best to join a faction as Liaison officers often have contacts within each and is the most usual way to disseminate information about an upcoming operation.
If you have anymore questions you can ask them on these forums, ask a faction member, or ask a Liaison officer for Org specific info, just remember Liaison Officers are generally playing In Character at all times. So just feel your way around and you'll soon get the gist of what goes on here and don't be afraid to ask questions.
Welcome to MxO!