Most racial differences will be reflected in the stats you see in the recruitment screen; it affects some starting stats and may have a couple other differences that manifest later from leveling. The only races with overlapping sets of classes are humans and shadowlings. The big difference between them in the same class is that shadowlings have 6 movement instead of 4, can fly, and have a small weakness to piercing damage. Human psy users can gain light armor mastery, while shadowling psy users can learn shadowport, a short-range self-teleport.

Things that units of the same species tend to have in common across classes, as far as my tired brain is aware:

Humans - sometimes gets light armor mastery (if they don't already get something better from their class)
Lissit - weaknesses to fire and cold, but increased movement and sometimes increased dodge
Shadowlings - increased movement, flying, small piercing weakness; sometimes gets shadowport skill at higher levels
Spriggats - increased movement, flying, large piercing weakness; has access to a few spriggat-only armor and weapons
Golems - high physical resistances, low movement; has access to some golem-only armor choices; has it's own pool of strength and weakness traits when generated
Spirits - very high physical resistances, flying, moderate mental weakness; no strength and weakness traits?

When you hover over the name of a unit's class in the recruitment screen, you are given a brief description of what the class is like. You can generally expect to get bigger and better versions of their starting skills as they level (more range, more damage, bigger effect area, etc.) as well as a few appropriate utility skills.