I second Advance Wars on the strategy front. And if you want some Real time Strategy, I've heard nothing but great htings about Robocalypse (It's the next game I'm getting)

Devil Survivor, Knights in the Nightmare and Final Fantasy IV.

I'll join the chorus recommending TWEWY. That game and Valkyrie Profile have taught me to like Square RPGs again after becoming disenchanted with Square's PS2 games.

I'd also recommend Civilization Revolutions, as long as the idea of a highly streamlined Civ doesn't repulse you. It keeps most of my favourite features from the main series (culture system, unit experience) and adds a few features that I wish the main series had (no tedious workers, simple army system, more rewards for exploration). My one gripe is that old units only upgrade in very special circumstances.

Oh, and you should grab any of the GBA Fire Emblem games you can find. The DS one is okay, maybe even a good way to get your feet wet if you're new to the franchise, but it's missing that special something...

3 originals:
Suikoden Tierkries, LOZ Phantom hourglass, Advance Wars/Super robotTaisen

3 remakes:
Front Mission, Chrono trigger, Fire Emblem

Now it's correct! Also add Luminous Arc  for a light SRPG experience with loads of sentimental effort in it ;) I can't recomend Black Sigil yet, as I just started playing it ;)

Any of these strategy games are good. Fire Emblem is your more typical SRPG and once your unit dies they die for good. Advance Wars you can build as many units as you like, but you have a small selection of COs to choose from.

Fire Emblem: Shadow Dragon (not a lot of story, but a good strategy game)
Fire Emblem (Best handheld Fire Emblem in my opinion, for GBA, can be found for ~10-20)
Fire Emblem: Sacred Stones (for GBA, can be found for ~10-20)
Advance Wars: Dual Strike (possibly the best Advance Wars in the series)
Advance Ward: Days of Ruin (darker storyline, very good, but most people like the others better)
Advance Wars (for GBA, can be found for ~10-20)
Advance Wars 2: Black Hole Rising (for GBA, can be found for ~10-20)

If you like rhythm games at all Elite Beat Agents is a great game. I picked it up for $5 in a whole collection of items for the DS and it was the best $5 in gaming I've ever spent.

Also, I highly recommend New Super Mario Bros for Platforming goodness, and Professor Layton and the Curious Village because it is one of the best puzzle games ever. It has a great story, presentation and selection of classic puzzles.