So, after a couple of years of borrowing my sister's DS, I bought my own today, with Pokemon Soul Silver. Main decision to buy my own DS was the Lufia remake, which I hope doesn't take too long to release here in the west. 

So far, I've played Legend of Zelda: Phantom Hourglass and Spirit Tracks, Final Fantasy III and IV, Metroid Prime Hunters, Black Sigil, Castlevania: Dawn of Sorrow, Portrait of Ruin and Order of Ecclesia, Pokemon Platinum, NSMBDS and Mario Kart DS. 

What other games do you fellow DS users recommend me? 

Get Pokemon Heart Gold (because everyone else bought Silver) I needz to trade pokemanz!

Otherwise, try out Phantasy Star 0, Nostalgia, Scribblenauts, Kingdom Hearts 358/2 days, the CoD games aren't bad, Moon needs people to actually buy it lol, and Dementum has a bit of a following.

Dragon Quest IV-V, Advance Wars DualStrike or days of Ruin, and Mario vs Donkey Kong.

Too late, I got Silver already

I've seen quite some people with Silver sigs, that's true xD Well.... it is the better version, I think

Thanks for your suggestions, I'll look into Moon and Dementum.  

Dragon Quest V is on my to buy list for next month, as it's my favourite Dragon Quest  

Are the Advance Wars games for the DS better than the GBA ones? 

If it's the story, you'll find Days of Ruin to be much better, otherwise they all play pretty much the same.
And while I'm posting you may also like: Chrono Trigger, Children of Mana and "The World ends with you".

The five must haves:

Dragon Quest V, Kirby's Canvas Curse, WarioWare Touched!, New Super Mario Bros Wii and Kirby's Super Star Ultra.

I would also consider one title a must have, but its an import title. Osu! Tatakae! Ouendan! If you have access to it, get it at all costs.

Other good games I would recommend:

Chrono Trigger DS, Dragon Quest Monsters: Joker, Retro Game Challenge, Kirby Squeak Squad, Henry Hatsworth in the Puzzling Adventure, n+ and WarioWare D.I.Y (though I have yet to play the last one).  As well as Lufia Estpolis, Okamiden, Dragon Quest IX and Dragon Quest Monsters: Joker 2 when they come out here.

Games which I'd recommend and think were not mentioned yet:

Adventure
Ace Attorney Series (Phoenix Wright, Apollo Justice)
Another Code
Broken Sword: Shadows of the Templars - Director's Cut
Hotel Dusk: Room 215
Secret Files: Tunguska
Secret Files 2: Puritas Cordis
Time Hollow
Runaway: A Twist of Fate (will be released this month afaik)

Strategy
Age of Empires: Mythologies or Age of Empires: The Age of Kings (note: NOT RTS on DS, but still great)
Anno 1701 or Dawn of Discovery (building sim)
Final Fantasy XII: Revenant Wings (RTS/RPG)
Fire Emblem: Shadow Dragon
Ninjatown (RTS, tower defense)
Sid Meier's Civilization Revolution

Puzzle
Bust-A-Move DS
Meteos
Picross 3D
Professor Layton and the Curious Village
24 All-Time Classics (aka Clubhouse Games in NA)
Tetris DS
Puzzle Quest: Challenge of the Warlords (puzzle/RPG mix)
Might & Magic: Clash of Heroes (puzzle/RPG mix)

Platform/Action
Soul Bubbles

RPG
Lunar Knights
Mario & Luigi: Bowser's Inside Story
Suikoden Tierkreis
The World Ends With You

I haven't ordered it yet, but I think Knights in the Nightmare is one of the most interesting games to import (the game was released in NA, but not in PAL region) - it looks phenomenal and atm I consider importing the special edition with the OST.

Other interesting imports from NA imo:
Etrian Odyssey II: Heroes of Lagaard
Glory of Heracles
Luminous Arc 2
Nostalgia
Sands of Destruction
Summon Night: Twin Age
The Wizard of Oz: Beyond The Yellow Brick Road

(Plenty of RPGs were not released in Europe and today's announcement of Q2 2010 DS release list for EU proves that this fact doesn't seem to change anytime soon).

@lestatdark, congrats. check my ds library , that might help u out on some gaming choices.

I would say Phoenix Wright and Professor Layton. You can have fun while working this brain of yours!