Tips for beating DMD mode:
1. Using the same save file, go through the game first on Normal, then Hard, and then DMD, making sure you get all the blue orb fragments possible (including the secret missions) so that you life bar is maxed out with 2 full bars. I suggest completing ALL the secret missions on Normal. Also, stock up on other items, namely Untouchables, since they are rare and expensive. I went through Normal and Hard using only 1 Untouchable and that was on Mundus on Hard.
2. Learn how to grenade-roll. This makes the grenade gun much more useful. Usually, after firing a grenade, there is a long recovery time. This can be cut short if you roll (hold R1 and press left or right). You can then fire another round and repeat over and over again.
3. Get all the moves for Ifrit and Alastor at most by the middle of Hard. Moves to buy first are Stinger (Alastor), Air Raid (Alastor), Air Hike (Alastor), Inferno (Ifrit), Kick13 (Ifrit) Rolling Blaze (Inferno). The rest will be very useful on Hard, but for Normal, you really only need Alastor moves.
4. Do not use Sparda. It has no Devil Trigger guage and if you do switch to it when you have a full gauge for Alastor or Ifrit, you will lose the DT you've built up if you switch back. You will want to use mostly Ifrit because Inferno, Meteor-lv.2, and Rolling Blaze are your friends.
5. The Bangle of Time is useful for defeating harder non-boss enemies like Death Scythe (these are everywhere on DMD). Use Ifrit and the grenade gun together. Alternate between grenade gun shots and kick13's because the kick13 comes out right after the grenade is fired, cancelling that long recovery time. I used this with a full 10-symbol Devil Gauge to instantly kill Nelo Angelo at full health inside Nightmare3 on DMD mode (Freeze him when he's taunting or "exhausted." Then kick and shoot him in the back.).
6. Become proficient in disposing of Shadows; they're everywhere on Hard and DMD. The easiest way to kill them without using items is to equip Alastor then DT and shoot them with the pistols very quickly. Once the core is exposed, use DT'ed Stingers. Then turn off your DT and avoid their death move. Once you are able to do this well, taking on 3 Shadows at the same time on DMD is not as hard as you would think.
6. On Hard or DMD, in the beginning of the FATE mission, head into the well to get the Sign of Chastity. There will be about 10 Blades onto you. Go and knock down the weak wall that leads to Sign of Chastity and jump up onto the platform. The Blades will follow. Use Ifrit and Rolling Blaze to keep jumping on them in that tight hallway, building up your DT at the same time as well as getting a Stylish combo. With at least 7-8 DT symbols filled, DT and do 3 Infernos on all the Blades. They should all die at about the same time, giving you about 600-700 red orbs. Now, go get the Sign of Chastity by jumping up through the hole in the ceiling. When you come back down through the hole the Blades come back! Repeat the process for killing them, jump up to where the Sign of Chastity was, jump back down and keep repeating until you are as rich as you want to be. This will also allow you to afford buying the remaining blue orbs, and Devil Stars, Vital Stars, Holy Waters and Untouchables which are so freakin' expensive on Hard and DMD mode. I suggest that you keep getting red orbs here until you have bought EVERYTHING you possibly can and keep doing it until you have about 40,000 red orbs if you are on DMD mode. I guarantee you will need to buy many Vital Stars and possibly Untouchables. This method is pretty fast and easy. I went from about 6,000 red orbs to about 23,000 in less than an hour.
7. On Hard and DMD, you don't have to face Griffon for the first time (after where you are supposed to get Ifrit). Just ignite the flame next to the door you are supposed to go through right when you get to it. Then when it changes to night time, go to where you were supposed to get Ifrit and an Untouchable will be there instead.
8. Five Untouchables are known to be scattered in the game on all difficulties in the same places: 1) outside where you fight Nelo Angelo the first time, on the roof of the castle to the left of the belfry; 2) when entering the ghost ship for the first time, check the treasure chest on the right; 3) when it turns into night time, go to where you first got Ifrit; 4) after you place the Staff of Hermes where it's supposed to go, go right into the picture of the bedroom, look on the roof of the bed; 5) go back to where you beat Nightmare the third time, look behind the stalagmites;
8. Check GAMEFAQS.COM and read the FAQs for Devil May Cry, particularly the "Ifrit DMD Boss/Enemy Guide" by TMurach, the "Secret Mission Guide" by JL Lee, and the "Blue Orb Piece Guide" by JL Lee.