This game will have 2 time periods that play and link together throughout the story. The first will be a prequel to the series following Sparda (Dante's father) in events that led up to his death (which haven't been revealed in the existing storyline). And the other will be set after Devil May Cry 2 (just because that was the last part of the story we've been given).
So the game starts with Sparda face to face with Mundus, and yes, the first fight of the game if the Emperor of Darkness himself. It is a tutorial battle that shows the player how to use some of the various mechanics in the game. Being the first battle it's not meant to be super difficult (unless you're playing on a high difficulty setting), and as per the existing lore Sparda defeats Mundus and seals him away. Little does he know this battle was being watched by a dark entity in shadows.
The game then transitions to the end of Devil May Cry 2, with Lucia hearing a motorcycle engine revving believe it to be Dante. This however turns out to be false, as a new character, who calls himself Kain appears looking for 'the Son of Sparda'. When Lucia claims she doesn't know where he is, Kain gets angry. As he swings for Lucia another motorbike emerges, this time it is Dante, who claims he's 'been through hell' to get here. A second boss battle begins; Kain vs Dante (which features a tutorial for Dante's moves which differ from his fathers). Although you win the fight, Kain doesn't go down. Kain suddenly gets hit by an RPG. This turns out to be from Lady who appears along with Trish. Kain is summoned back by his master (the dark entity from the intro fight). Kain tells Dante and Co. he will return, Dante quips 'call me' and Kain grunts while driving off into the distance.
I don't have a full plot to this game at the moment, but it would include Nero and Vergil in its plot somewhere. The main thing I was going for is to follow a storyline for Sparda post-Mundus and have it tie in to current events for Dante. Also the dark entity who was watching the intro fight would be the 'big bad' who commands Kain and would be the final boss of the game.
Also there's rumours of Devil May Cry 5 being announced at E3, hopefully that's true.