Some ideas for fighters:
-fighters get two backgrounds
-when performing a check in relation to their background are considered to have proficiency in the skill if they do not already have it, ie: a fighter with the noble background is considered to have diplomacy(?) when talking to a king or insight on the motives of the other participants at a high class ball, a sailor has acrobatics to climb the rigging of a ship in the middle of a battle or insight?perception? (proficiency on something or other) when searching for information at the docks, the urchin fighter has proficiency to pick a pocket with slight of hand or blending into the crowd with stealth
-the fighter can spend the duration of a rest period to train another character to be proficient in any skill or proficiency they possess themselves, a short rest they are proficient for 8 hours, a long rest 24 hours
-the fighter has a climb (and swim?) speed of half their standard movement speed
-when taking a feat that does not already provide an ASI the fighter can add +1 to a relevant ability score to the feat they’re taking
Combat focused things
-the size/number of damage die a weapon deals is increased by 1 when used by the fighter (greater die size increase at later levels)
-when under half health (and conscious) the fighter recovered 1d4 health at the start of their turn in battle (1d6 at higher levels)
-the fighter can treat every weapon as if it has the Thrown trait