As Xavier warns Wolverine that he was a very different man in the 70s, we see his younger self entering Cerebro and blowing a thick layer of dust off of the helmet. My personal theory on this is that Beast and Xavier built the new Cerebro after the events of First Class but Xavier succumbed to post-traumatic stress disorder later down the line (hence him not being as well groomed and suave looking as we've seen him before) and left it neglected for years until Wolverine's arrival. Which is why the helmet is so dusty.
"Lead me. Guide me. Be patient with me." Xavier asks Wolverine to take on the role of his mentor in an interesting role reversal of the first three films. Wolverine's response is a typically snarky "Patience isn't one of my strong suits". We know, Wolvey. We know. We also see Xavier probing Wolverine's mind again (or should that be for the first time), which is quite an interesting echo of the first film and probably the turning point where they start working together.
After that, it's a series of fast action beats as we see Wolverine being transported through time followed by what looks like Colossus running away from an attack on the temple (Here's hoping that he gets more to do this time than standing around and punching things). We then see Young Magneto floating up into the air, possibly as a show of force or as part of some kind of politically motivated attack. We also get some quick shots of what looks like a riot, which is to be expected as Days Of Future Past will feature a lot of anti-mutant sentiment stirred up by several different factions as well as including the general political unrest of the 1970s.
There's also a slightly longer shot of Mystique in her Raven Darkholme disguise, possibly bringing up her appearance issues from First Class again and showing her temporarily rejecting the "mutant and proud" belief. Another reason might be that she's using it for some kind of espionage work for the Brotherhood.
Richard Nixon enters the picture as his security staff open up a hatch in the floor of the Oval Office that leads to a series of catacombs and an underground bunker, and escort him, Bolivar Trask, and a number of other people there. Maybe Magneto attacks the White House to kickstart the Mutant revolution, or Trask's Sentinel robots go rogue and attack indiscriminately.
And, of course, it wouldn't be an X-Men film without somebody attacking Wolverine followed by him healing, going into a rage and killing everything in sight. As for who the guys shooting him are, since there's nothing that really identifies them, I'm drawing a blank. Maybe they're anti-Mutant vigilantes.