I'm wondering if the Fighters have spells, or if they have moves they can use besides just attacking?
Yes. Fighters who choose Eldritch Knight as their Martial Archetype gain access to spellcasting.
At 3rd level, a fighter chooses a Martial Archetype. The PHB (Player's Handbook) contains the options Champion, Battle Master, and Eldritch Knight.
In short, champions are athletic and they specialize in hitting things. Battle Masters are skilled at maneuvering their enemies around the battlefield, such as by pushing or tripping. Eldritch Knights can cast spells, in addition to their regular weapon attacks.
The Eldritch Knight uses Intelligence as their primary spellcasting ability score. They are very limited to compared to full spellcasters, though. They eventually gain access to 2nd, 3rd, and 4th level spell slots, but their spell progression is much slower than full spellcasters.
More information about the class is available in the PHB, such as the list of Wizard spells (which may become available to an Eldritch Knight upon leveling up). Even a quick Google search for "5e fighter" will lead to websites with the details on all of the Fighter class features.