Yes to all of the suggestions so far. I never thought I'd be saying this, but if you have a PSP you've got a lot of the good ones available. The original FFT was ported and retranslated, Tactics Ogre's on there, Disgaea's ported there (the subtitle of this game gives a hint at the humor of it... the original subtitle was Hour of Darkness, the PSP version is Afternoon of Darkness). A fair number are on the DS, though, and like Meta said, you can get most of the originals for the PS/PS2. Those are the main three series I typically hear throw around when people are talking about good SRPGs. I haven't played Super Robot Wars, but this isn't the first time it's been brought up, so it's gotta be good.
Edit: restructuring based on OP clarification:
The FFT series: good for a solid class system where you can make any character good at most things, eventually. The means to do this are somewhat opaque, though. One comment about the FFT "series" though: the original is very different tone and story-wise from the Advance games. I like them both, but for different reasons, so if you're looking for a serious "old school" kind of feel, go with the original. If you're looking for a still fun but lighter-hearted story, go with the FFTA games.
The Disgaea series: good for a fair amount of over the top mischief and mayhem with a good dose of humor, plus character customization that goes beyond a "class system." You can sink a LOT of time into Disgaea if you want. I'd say this game's system is the closest to FFT's base, but it builds on it and adds some mostly-optional complexity.
I know less about the tactics ogre systems, but it has a more old-school feel to it as well. I think the PSP one is more polished than the advance one (and I think the PSP one's a remake of a super famicon/PS1 game), but they're pretty fun. I remember getting a little frustrated with the balance of how the battle system worked, though (I felt like my characters were missing a lot and not doing much damage and the range of archers is extended by being higher up, but you don't know by how much since the base range is shown but any modifications aren't. I think this archer range complaint applies to the original FFT as well, I just never used them in that game), and I feel like the class system is even more opaque than the other two since it relies on getting a character's stats to certain levels instead of just having a certain class or class combination at certain levels. If you're playing with an FAQ/cheat sheet for that aspect of the game like I typically do it's less of an issue, but eh.
Fire Emblem's good too, but if I remember right there's very little character customization; more of the emphasis is on the in-battle strategy aspect.