Dungeons and Dragons Online

What style / class should I play?

The classes for DDO are broken into three groups called "styles".
Melee characters engage in up-close hand-to-hand combat. Choose this style if you want to attack with weapons instead of magic.
Spell casters harness the power of magic to assist allies or attack their enemies from a distance. Choose this style if you want to use magic more than weapons.
Specialists can excel in scouting, ranged combat, trap disarming and more. Choose this style if you want a more specialized role.
DDO suggests that you go with Melee / Fighter / Vanguard Warrior.

If you want to play a caster, they suggest Cleric, and for specialists they suggest Ranger.

They also rate each class according to soloing ability:
Melee Fighter Good
Barbarian Good
Paladin Very Good
Monk Very Good
Spell Sorcerer Challenging
Cleric Very Good
Wizard Challenging
Favored SoulGood
SpecialistRanger Good
Rogue Challenging
Bard Challenging
The game supports multiclassing, if you are willing to do a little more work, you can get some very nice rewards.

I will make two specific suggestions, but as you play, you will probably realize there are other paths that are interesting to you, and you will roll up some other characters to try them out. You should definitely do this, experimentation and learning are always good.

Warpriest

I suggest that you start off with a warrior priest.

The game has a pure cleric version of this, Spell / Cleric / Warpriest of Siberys, but the version I am suggesting is actually a multiclass character.

First, you start off with a level of melee character to gain access to martial weapons, after that you level up as a cleric who can also do serious melee damage.

There are four classes that give you access to martial weapons. Which martial class to start off with? Any of them will do, but each has pros and cons:

Barbarian
Pros: Fast Movement, Rage, Hit points
Fast movement is easy match or exceed with gear. Rage can (rarely) be useful to open doors or in combat.

Fighter
Pros: Bonus fighter feat.

Paladin
Pros: Aura of Good, Smite Evil.
The Aura of Good is a fantastic bonus, boosting AC and saves.

Ranger
Pros: Favored Enemy, Wild Empathy, Bow Strength.
Bow strength is a great addition to ranged damage.