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Fighters are heroes who go about hitting evil things for a living. They're generally good at surviving long enough to brag about it afterwards.
Due to his lack of specialisation, the Fighter works well with any race. The only specialised race choice is Goblin, combining the Veteran trait with the ability to gain even more XP from conversion for extremely rapid levelling. However, it is essential to leverage the level-ups properly, since a high-level Goblin Fighter has nothing going for him compared to other characters of the same level. With other races, the fighter is a more laid-back experience.
The fighter levels up significantly faster than other classes, which has two huge benefits. Being a higher level he is more powerful and dangerous, but it also benefits more often from the full-heal and mana-restore you gain when you level up. This extra leeway in terms of experience gain and leveling up makes the Fighter a competitive class. His other class features are largely inconsequential. Pit Dog is a one-time event, and basically equivalent to one use of the CYDSTEPP glyph. Instinct is completely useless, since fighters want to avoid fighting monsters that are equal or lower level than themselves if at all possible, saving them for level-up catapults later in the game.
Although he's presented as a Fighter, this character is actually equally well-suited to being a spellcaster. He has no class features that help his melee combat skills, and largely comes across as a tabula rasa that can be played any way you want. It's always a good idea to keep an open mind with the fighter and be as flexible as possible based on what items and glyphs you find.
This is the first class you are presented with when you begin Desktop Dungeons, and he functions very well as a beginner class, and also presents a strong foil to the specialized classes such as Warlord and Assassin that are unlocked later.
As for deities, Tikki Tooki is a natural choice, as he favours characters who reach a high level quickly, and a Fighter with Tikki's Edge will zoom through the character levels. Glowing Guardian also works well: the Fighter's XP bonuses synergise very well with the Humility boon, and the Fighter also has enough spare XP to make liberal use of Absolution. The Fighter works well enough with any other deity as well, so for the most part you should adapt your worship strategy to the needs of the dungeon.