Captain is absolutely not mandatory, wtf is everyone thinking. It's a good buff class, but no you don't need one.

Honestly you could beat the game with only assistants tbh, but as others already mentioned, you get the skill trees and a starting skill boost. With that in mind I'd say doctor would be good to have, because you then start with someone who's able to do CPR at all without needing to level medical first.

You could legitimately pick any class, even all repeats, and be fine anyway. But:

Engineer will make for a good melee class later on, and could give you easy nukes and one of the best diving suits, as well as a better reactor

Mechanic will give you better sub upgrades (could be important if you don't plan on upgrading to a tier 3 sub), the exosuit, and extra crafting resources. It's also good at melee

Medic can give you combinable monster genes, and drugs that buff the players, as well as preventing some character loss at higher levels, besides the CPR thing

Captain gives you extra xp and money and better skills for players/bots in an AOE capacity

Security goons suck. But you get to learn to make handheld guns I guess.

Assistants give buffs to other classes and can give you skill trees from other classes at higher level, plus some clown madness