Okay here's some tips.
First and foremost, have all of your characters learn the mantra line that's their weakness. Why? Because you learn abilities and spells that negate that weakness. When you have finally mastered the entire row, you earn an ability that allows you void and null your weakness if it is ever used on that character.
Let's use Seraph as a good example. Seraph's weakness (if I remember) is fire. For strategy, it would wise to cover up that said weakness as fast as possible, so not to get raped, and also so you can gain the upper hand when enemies are dumb enough to attempt to exploit your weakness. By learning Void Fire and Null Fire or Resist Fire and whatever gobblety gook, you can gain the upper hand when some one uses that spell against Seraph. The first thing you should always do when playing a DDS, or SMT game in general, is cover your weakness. If you have a fire based demon, make it so he can some how learn an ice based protection spell or ability.
Secondly, it's wise - and common sense - to learn your strengths. Let's continue with our Seraph example from before. His innate spell is Bufu aka ice. By learning stronger ice spells, you can tear shit up more easily. What's more, as you go along the ice mantra sets, you'll learn things like Ice Amp and Ice Boost, which propel your ice magic power. Very handy.
Thirdly, and this is just as important as covering your weaknesses, learn strong devour and physical abilities. Alongside weakness mantra line, you should totally look into learning higher level devour and physical attacks. There are many attacks that devour and cause physical damage at the same time, so you don't have to keep hitting an enemy until he's weak and hope your don't kill him and miss your chance to gain Atma Points. Remember, the faster you kick ass, and manage to devour enemies the more AP you get. When you get to the Samsara sewers, I'll tell you about a lil' trick that allows you to get so much AP you can master an entire mantra in one battle.
Use weaknesses! This is key. You will not get far without using them.
Bullets, and human form, are often thought to be a waste since they're considerably weaker than demon form, but remember that humans cannot come under the influence by status effects and some enemies go down way easier with a mere bullet rather than the claw of a demon. Most of those guys are flying enemies, so watch out for them.
Key abilities to learn (ones you should emphasize on getting, if not asap, eventually. use the map on gamefaqs to find out their location and how much they cost):
Weaknesses and their resists - FIRST THING YOU SHOULD GET.
Strengths and boosts + amps - SECOND THING YOU SHOULD GET.
AP Divide - GET THIS IMMEDIATELY. As the name implies, you can divide the ap you get between party members, so you inactives can get shit. This is handy, not because DDS forces you to use certain members at certain times, but because you should really look into mastering a good portion of the grid with most characters before beating the game. If I remember, mastering certain parts (or even, the ENTIRE mantra grid) gets you bonuses in DDS2 at the beginning of DDS2)
Ragnarok - Amazing physical ability
Phys Resist - Will be very handy near the end of the game, along with Phys Drain.
Last Meal - Amazing Devour ability
Blood Bath
Iron Stomach - GET THIS IMMEDIATELY. It basically makes it so you don't get stomach aches after eating your enemies.
Double Slash
Counter - GET THIS IMMEDIATELY. When an enemy hits you, you them back. Extremely important. Your enemies even do this so it's best to even out the playing field.
Dia - GET THIS IMMEDIATELY. Dia mantra line are healing spells. Very very useful, obviously! Likewise, the sister set of healing mantra spells are spells that cure and prevent status effects.
Mudo - GET THIS IMMEDIATELY. Death spell. It's not very useful in the later part of the game because enemies often void, resist, or null it, but early on it's handy, along with it's more powerful version Mudoon - which can kill all enemies.
Hama - Get this about half way through the game.
Void Expel - Get this about half way through the game.
Rakunda - VERY IMPORTANT. Buffs are VERY important in the later part of the game.
Dekaja -

Dekunda -

Marin Karin -

Tarukaja -

Pulinpa -

Makajamon -

Calm Death -

Mutual Karma - GET THIS. GET IT.
The others, that are on the far side are unlocked after killing secret bosses. They aren't necessary but damn are they awesome! They are expensive as FUCK though!
I cannot stress enough how awesome and how important the Kaja's and Kunda's are in later part of the game!
Finally, combos. These can use up an entire turn, but they are VERY handy if you do not have access to the spells or have enough AP to buy them. You can get some higher level magic and stuff earlier using this tactic. As an example, when you have two characters in your party who have learned Dia, simply go to the combo section and they'll use Media. Media is the second strongest healing spell. At the expense of wasting entire turns, you can use higher level stuff early on. Make use of this advantage.