The only game in the series I have played is Devil May Cry 4 (referred to as DMC4 from now on). I thought that was an average game with plenty of limitations so only awarded it a 6/10. Although some fans were annoyed at this DMC reboot, I was impartial and would judge this game on its own merits.
The combat remains much the same, so you move with the left thumb-stick, jump with A, shoot with X, sword attack with Y, and B launches enemies into the air. Soon into the game, you get your first Angel/Demon weapons; an axe and a scythe which can be used by holding down the Right Trigger or Left Trigger respectively. Additional Angel/Demon weapons are found later into the game and can be switched out using the D-Pad. You also get a grappling hook which is used with a combination of the Triggers and the X button. This can pull enemies towards Dante, or pull Dante towards enemies. Outside battle, this is used as a grappling hook in specific areas. You can't lock onto enemies, but the game does a decent job of targeting so isn't a problem.
You are encouraged to launch enemies into the air and keep them from touching the ground, and to avoid taking damage. The combat has been vastly enhanced from DMC4 so there's a larger move-list. The simple sword combos you do start out with are YYYY, or YYPauseYYY, so there's nothing too complicated, but the complications come from the enemy types. It gets a bit more hectic when there's multiple enemies, but some are shielded or weak/resistant to either Demon or Angel weapons.
The game very much reminds me very much of Bayonetta since it contains hack & slash game-play and extremely over-the-top action. The game is fairly linear, but gives you a bit more freedom compared to DMC4 which usually constantly locked you in small rooms until you defeated the enemies. The camera is far better in this game too, often giving you control over the camera, rather than the locked and disorienting angles. The cut-scenes are usually just at the start and end of the levels, rather than giving you constant disruptions; so that's another improvement.
You collect items in a similar way to DMC4. There's red orbs which are dropped from enemies which can be used to purchase items and health upgrades. The white orbs are awarded based on your performance which are used to unlock new moves. You can get a refund on moves if you decide you don't like them. There are hidden keys to find which unlock doors within the levels which take you to challenge rooms. These challenges have a variety of ideas, including time-trial platforming, clearing out multiple enemies with a certain type of attack, or taking down a powerful enemy with a strict time-limit.
Dante is a hybrid of angel and demon; known as Nephilim. He meets a mysterious girl called Kat who explains that she can see into Limbo which is basically a parallel universe where demons reside. Dante will often get dragged into Limbo and have to deal with hoards of demons. The game takes advantage of the Limbo idea by creating a world that is visually stunning. Often, the scenery will morph in front of you, with roads being ripped apart to form platforms. Dante teams up with Kat, and meets Virgil who is soon revealed to be his brother and they plot to take down Mundus. He is a powerful demon who is essentially ruling over the humans after brainwashing them via the TV network and with spiked health drinks. There's plenty of swearing in the game, most of the time it feels forced just to make the game seem more edgy.
DMC is a massive improvement over DMC4. It's a great action game, with a wonderful aesthetic.