Sisyphus and the Meme Bunch set out to save the world.
Spongylus the Hero (Personality: Tough)
Following Dragon Quest norms, the protagonist is a bit of a jack-of-all-trades - Good equipment choices, solid physical stats, mediocre spellcasting stats (though they do get some very useful utility spells) and not very fast. You also can't change out of the Hero class, so that's what you're stuck with. You can tweak your character's stats a bit with the Personality system in the remakes, at least, so I went with one that plays best to his strengths - Tough gives a whopping 140% Vitality growth and 115% Strength, while the ones that don't serve him as well get lower growth (Agility 90%, Int 80%, Luck 70%).
Azul the Thief (Personality: Silly)
The Thief class is new to the remakes, and fits about where you'd expect - high agility, pretty decent equipment choices (they can wield Whips and Boomerangs - quite handy for taking down monsters quickly) and a handful of special abilities that help them track down treasures and evade enemies. Their biggest downside is their lousy HP growth, though this isn't a big problem early on - Agility adds to Defense in this game, so they can still take a punch pretty well. It gets to be a liability later on when enemies start casting offensive magic a lot more (which ignores Defense entirely), so I plan to make her a Sage when the opportunity arises. In the meantime, the Silly personality grants a 120% bonus to Agility and 115% to Intelligence, so she'll have more MP to put to better use when that day comes. Silly does give a slight penalty (95%) to Vitality growth, but that shouldn't be too noticeable in the long run.
I should also note that Thieves are actually detrimental to collecting Monster medals, so if you want to complete that bit of bonus content, you should probably switch out your Thief later on (or just start with another class instead - Dealer is a good choice for their extra cash gains).
Torgo the Cleric (Personality: Smart/Sharp)
It's an RPG, so of course having a dedicated healer around is basically mandatory. While not as strictly required in the remakes, having one around for free on-the-fly healing is still a good idea, especially in the early game when money is pretty scarce. For Personality, I went with Smart (130% Int, 95% Str), though I accidentally overwrote it with Sharp later (120% Agility, 140% Int, 80% Vit); hopefully that HP loss won't come back to bite me too much. Worst case I can overwrite it with something more fitting later, like Fearless or Kindly.
Trogdor the Jester (Personality: Sexy)
Jester, as the guy at the bar tells you, is easily the worst class in the game - they randomly waste turns doing silly things, and their stats are generally crappy save for pretty good MP growth (rather questionable considering they get only one spell, and it costs 0 MP to use) and amazing Luck (shame Luck isn't really very good at all...). So, why bother with one? Well, once the ability to class-change becomes available at the Shrine of Dhama, they can become a Sage without having to spend a Zen Scroll. You still have to hit level 20 first, but that's a minimal concern as Jesters level up surprisingly fast; they'll usually be a level or two (or three) above the rest of your party. Their equipment choices are surprisingly decent, too - they can use Whips and Boomerangs and most mid-to-light pieces of armor. So while they are objectively the worst class, at least they're not dead weight.
Sexy is probably the single best stat-boosting personality in the game and exclusive to female characters. Check this out:
Strength: 110%, Agility: 120%, Vitality: 105%, Int: 115%, Luck: 120%
It's not the best Personality for any one stat, but considering Sages are designed to do a little of everything, it's a good fit for any character you plan to make into one.
And on that note:
Coming Soon: The Sage Class
Sages are basically a combination of Clerics and Wizards, combining the high MP and Intelligence of the latter with the solid (if not amazing) physical stats of the former, and they get all the spells of both as well as some equipment choices that neither of the base classes get. Really, their only downsides are that you need a Book of Zen or a Jester to become one, and that they level up slower than any other class - you'll need about 1.5x as much experience to gain a level as any other class.