While it's true Warriors possess greater strength and vitality than other classes, they are still quite pretty vulnerable unarmed or when wielding low-grade weapons and are largely dependant on the equipment they have. Unlike the other classes who can attack effectively from distance, for Warrior's their only real method of killing monsters is to enter into hand-to-hand combat – an activity which relies on the existence of decent equipment in the character's inventory.
Warriors are best in the early and middle levels of the game and against tough solitary monsters. As they gain levels and obtain better equipment, Warriors can hack through waves of monsters relatively easily when close up, but often struggles in higher levels where there is a greater incidence of spell-wielding and missile-equipped monsters. When faced with a group of monsters, try to lure some away from the group in order to whittle down numbers gradually rather than wading into the middle of a horde.
The best way to play a warrior is to take advantage of his skills with weapons and strength by wearing heavy armor and shield, and wielding a powerful weapon – if magical, so much the better. Axes generally do more damage than other weapons, but remember axes in Diablo require two hands, so you can't use a shield. Blocking, especially in higher levels really comes into it's own and with some shields giving Fast Blocking abilities, it may be better to invest in a solid one-handed weapon and use a strong shield if you want to progress.
While having a Warrior with some magical talent would be beneficial, it shouldn't be hung upon too much. Spells to concentrate on learning are Stone Curse, Chain Lightning, Firewall, Healing and Teleport. Spells are learned through books which can be found throughout the dungeon, although occasionally can be bought from Adria.
Remember, however, you're a Warrior first and foremost, so you'll probably be better off concentrating on physical attacks to begin with than developing magic. Item characteristics that help improve Warrior characters are weapons with fast attack bonuses and faster hit recovery.
Dealing as much damage as possible isn't the key to playing a successful Warrior; the ability hit opponents and to soak up huge amounts of damage is. After all, if you can't hit them, you can't inflict damage and vice versa. Concentrate instead on building up your character's armor class and resistances to ward against monsters whose primary attack is missiles or ranged spells.