This may sound a little odd coming from the guy who started the "Rank the 3.5e DMG PrCs" thread, but I can't really rank the usefulness of the core classes. With the prestige classes, you had the core classes you could compare them to. With the core classes, though, each one is designed to fill a specific role that sets it apart from the others (well, the wizard and sorcerer fit pretty much the same role, but other than that the classes are distinctive). Here are my thoughts...
Barbarian: I've always had a soft spot for barbarians, since I'm a huge fan of the Conan stories by Robert E. Howard. In 3.5e, barbarians are absolute terrors in melee combat, posessing high hit points, powerful attacks, and the ability to take a lot of punishment before falling.
Bard: A good support character if you've already got the four "required" party members (A frontline warrior, a rogue, a cleric, and a sorcerer or wizard). His social skills can make relations with NPCs a lot easier, and his bardic music and spells come in handy as well.
Cleric: The cleric has always been the backbone of a D&D party. He posesses healing and protective magics that are virtually required for party survival, the ability to turn undead, a few nice damage-dealing spells, and he's a decent melee combatant as well.
Druid: While some might dismiss this class as a "cleric-lite", I've actually seen druids used to great effect in D&D sessions... Very few monsters expect a nature-loving half-elf to suddenly morph into a raging polar bear.
Fighter: While he may not possess the raw melee power and toughness of the barbarian, or the holy powers and celestial mount of the paladin, the fighter remains a tough, competent brawler that makes a find addition to any party. The fighter is also the most versitile class in the game; depending on how you distribute his ability scores and feats, he could be a dedicate marksman with the bow, a greataxe-weilding melee juggernaut, a quick and graceful swashbuckler, etc.
Monk: Even the monk has his niche, even though it's harder to pin down than most. Monks are fantastic at mage-killing; able to rapidly close the distance between themselves and their target, and then grappling them and pummeling them to death.
Paladin: Another class that looks weak on paper, but in actual gameplay can prove to be very useful. A high-chrisma paladin has excellent saving throws, good healing power from Lay on Hands, and the power to smite evil with the best of 'em. His immunity to disease and fear is also very useful.
Ranger: Probably the most controversial D&D class of all time, simply because everybody seems to have their own definition of what a ranger is and what abilites they should have. That being said, I love the 3.5e ranger (the 3.0 ranger was useless, I always just used Monte Cook's variant ranger). The 3.5e ranger is more versitile than ever, and like the bard makes a wonderful "fifth party member."
Rogue: An absolutely indespensible class for dungeon exploration, due to his ability to pick locks and disable traps. The rogue is also capable of good damage when he gets into flanking position and starts dealing sneak attack damage. Very useful.
Sorcerer & Wizard: What can I say about these classes? A virtual must-have for any group. From
magic missile all the way to
wail of the banshee, these guys bring a lot to the party.