Barbarian: Human, Dwarf, Half-Orc. Humans get more skills, and the feat (this will be a repetitive theme, and applies for all human entries here on in). Dwarf gets CON bonus (very important for Rage), CHA negative (who cares?), and with the new movement rules, a Dwarven Barbarian in Medium Armor moves just as fast as any other race. Half-Orcs get STR bonus (good), and an INT and CHA penalty (who cares?).
Bard: Human, Half-Elf, Gnome. Gnome because of the bonus cantrips and bonus to illusion DC's (many Bard spells are illusionary). The Gnome's bonus to CON helps a lot, and the STR penalty doesn't hurt the Bard very much. Half-Elves get a bonus to Diplomacy now, and that's a major skill for Bards.
Cleric: Human, Gold Dwarf (FRCS). Gold Dwarves (+2 CON, -2 DEX) aren't bad, and the new movement rules for Dwarves allow them to be just as mobile as other characters in heavy armor.
Druid: Human, Gnome. I'd suggest putting your lowest stats in STR and DEX, because you can overcome these deficiencies with Wildshape. Gnomes get a bonus to CON (you always need HP), and the STR penalty doesn't really hurt them.
Fighter: Human, Wood Elf, Half-Orc, Dwarf. Wood Elves because of the Stat modifiers (you need to watch that CON, but the INT penalty shouldn't matter, since there aren't any good Fighter skills anyway, and you get bonuses to two physical stats). Half-Orcs (+2 STR, -2 INT & CHA), for similar reasons as the Wood Elf, but the Half-Orc doesn't get a CON negative, and darkvision is nice. Dwarves now move as fast as any tall race in armor (and your fighter should be wearing armor!), get a CON bunus, and a CHA penalty -- sounds good to me. One thing to note: Fighters are very customizeable (is that a word?), and a case could be made for any race to be a very good fighter, it it's own regard.
Monk: Human. Skill points are scarce, and you want as many as you can get. The extra feat really helps too.
Paladin: Human, Gold Dwarf. See Cleric.
Ranger: Human, Elf, Wood Elf. Elves and Wood Elves both have CON penalties, but the DEX bonus helps a lot since they need to wear light armor, and the archery combat path is heavily reliant on DEX. Watch your CON, and make sure you rolled a high INT if you're going with Wood Elf (you're going to want skills).
Rogue: Human, Halfling, Gnome, Elf. The DEX bonus the Elves and Halfling offer really helps a lot, as DEX is the most important stat for Rogues. Halflings get bonuses to a few really important Rogue skills. The penalty to STR that Gnomes and Halflings get shouldn't matter too much, since you want your damage to be sneak attack damage anyway, and that outweighs a +/-1 anyday. Lowlight vision of the Gnome and Elf is really nice for people who want to sneak around in the dark. As with Fighters, Rogues are extremely customizeable, and you can go a lot of different directions with this class.
Sorcerer: Human, Gnome, Halfling, Elf. I like small spellcasters. The spells are just as powerful, STR penalties don't matter, and even with a reduced speed, your movement is just as good as most party members (and faster than the heavy-armored ones). The small guys get an AC bonus too (and you take it where you can get it with spellcasters!). The Halfling gets a DEX bonus (more AC; better initiative), and a bonus on all saving throws. The Gnome gets a higher CON (HP's are even more important than AC), gains a few extra cantrips (more spells are your friend!), and a bonus on all illusion spells. WOW! I think the Gnome really is the best race for Sorcerer, but the Human and Halfling really do need to be mentioned. If someone were looking to play an Illusionist, I'd steer them towards a Gnome Sorcerer instead, but I guess I'm a bit biased. An Elf gets some nice added Weapon Proficiencies (a longbow is nice for when you run out of spells), but the CON penalty would have me steer clear.
Wizard: Human, Gnome, Halfling, Elf, Dwarf. See Sorcerer. Dwarf was added because CHA isn't needed for a Wizard, and a CON bonus really helps anybody with 1d4 HD out. As noted above, I'd rather go with a small sized Wizard than a Human, Elf, or Dwarf.
~munchy krunchy out
ON EDIT: I know you said "Core Races only," so ignore the parts I wrote about Wood Elves and Gold Dwarves. Many DM's will allow these non-PHB races, so I'm not going to edit them out.
Spelling errors fixed too.