It depends entirely on the grade of cortex. The old ikrpg
d20 materiel went into it in great depth.
The most basic ones are incapable of much more than 'single' instructions. For example, they could follow the command 'go to the gate with this trailer and wait', or 'carry this crate to the warehouse'. Anything more than that and they'll hiss and clank and fumble about in confusion.
More advanced models can do more than a single set of instructions. For example, they could follow the command 'carry these crates from the ship hold to the elevator, then across the boarding plank to the warehouse and stack them on the ground floor, with the biggest ones on the bottom'.
Obviously, the latter is far more expensive to build, buy, and maintain.
The military jacks are the highest grade cortex. Generally, the best analogy is on their own they're as smart as a decently trained dog. But they're military. All their skills involve stomping things, punching things, shooting things and hitting them. So they're better, but in a more limited set of skills. You couldn't, say, take a jack cortex and put it driving a tugboat - it would try to charge and slam every other ship.
With warcasters, you have a thing called an 'imprint'. When a warcaster bonds with a jack, and uses it, and their magic flows through a jack via their mental connection, and via mental commands, traces of their personslity get imprinted on a cortex. Do it more and more, and this imprint becomes more and more pronounced. If a warcaster constantly tells his jacks to be cautious and on guard, test will become their default 'stance'. Some jacks be one very protective like beast 09. Others become extraordinarily bloodthirsty,like drago. Others develop more unusual quirks. Rockinante, for example is friendly to kids and quite gentle with them. It's not a 'true' personslity, merely an echo of a warcasters mental programming. These traits can be dangerous, which is why midst cortexes tend to be wiped after use. For example, would you ever want to keep a jack the butcher has had under his control unwiped? Unless of course what a warjack 'knows' amounts to valuable combat protocols, then the decision might be made to keep its cortex as it is. Ol rowdy for example has very valuable combat experience and has developed a very 'eager' personality. Imagine a Labrador that always wants to go out on adventures chasing things. His combat experience was deemed extremely valuable and of great utility to new warcasters. His 'ornery' personslity likewise is seen as a challenge to control as well.
But as mentioned, as smarts go, think of a dog, and a warjack will be somewhat similar.