Really basic guide for most classes
Stat points
Enough in strength for your gear (usually 60-80 base points, depending on gear)
Usually either nothing in dexterity or enough to hit 75% chance to block if you have a shield. There's a formula on the Arreat Summit to calculate how much dex you need for 75%. Also, bow-wielding Amazons need lots of dex, although they're difficult and gear-dependent.
Very little to nothing in energy (even a Sorceress doesn't need tons of energy; you can make up for low mana with items like Frostburn gloves and lots of mana potions)
Everything else in vitality
Skill points
Pick four skills that work well together. Max them out. Read their descriptions to look for synergy bonuses (which is where points in one skill add damage to another skill). Put one point into as many passive bonus skills as you can, and then start working on a fifth skill once the other four are maxed.
Two really popular skill builds are these:
Meteorb (meteor + frozen orb) sorceress
20 Fire Ball
20 Meteor
20 Frozen Orb
20 Fire Mastery
Rest in Cold Mastery
1+ Teleport
Skeleton necromancer
20 Raise Skeleton
20 Skeleton Mastery
20 Corpse Explosion
1+ Amplify Damage
1+ Clay Golem
1+ Golem Mastery
1+ Revive
1+ Skeleton Mage
1 Decrepify
I particularly like that necromancer because you have so much freedom to place skill points wherever you want. By the time you have decent gear, you will have a wall of 20+ summons to protect you from everything. With curses, you can get away with just a few points and rely on gear to increase your skill level. And Corpse Explosion is fun.
Gear
Sorceresses, Necromancers, Trap-Assassins, Lightning Amazons, and any other class that relies on magic spells to do damage will benefit immensely from gear that grants extra skill points. Magic generally increases in damage exponentially with more skill points, so even just one or two extra points can make you do a lot more damage. These classes also need lots of mana to keep their attacks going.
Melee classes don't really need +skill gear. Extra points in combat skills either add damage linearly or actually decrease in effect with more points, so you have to get high damage gear to do well. What melee classes absolutely need more than anything else is life and mana leech. X% life/mana stolen per hit is essential. Other helpful things for melee classes are % damage reduction (a few items offer this), +damage charms, resistances, and a backup attack for physical immune monsters.