Most important lessons -- outside of litterbox, which luckily comes naturally to most cats with just a bit of encouragement -- are practical ones.
1) Some surfaces are Not For Kitty. Food prep and serving surfaces are off limits in my house. One of the bookshelves likewise because it usually isn't stable enough. The cats learned this, after being told "NO! Down!" A few times. They tested periodically over the next year or two to make sure the rule was still in force -- there's a nice sunny spot on the dining room table so I can't entirely blame them -- but at this point they know and follow the rule. At least, I haven't caught them in a few years, which is all I can reasonably ask.
2) "This is a scratching-allowed surface, that isn't." Scratching stretches the claw muscles and helps shed worn layers of claw, as well as asserting "This is part of my territory", so cats do need an acceptable place to scratch -- scratching post or corrugated cardboard or something similar. But they also have to understand that the good furniture is Yours, and they can't scratch on it. Again, telling them No firmly is often enough; if necessary pick them up and move them to the acceptable surface. Mine will sometimes now pause in mid-stretch and look at me to check that it's ok to finish the scratch.
3) Electric cords are not cat toys. Give them other toys to play with, and tell them no when they go after something they shouldn't. They'll figure out what you're trying to teach them as long as you're consistent and firm about it.
4) House policies re sleeping people. Mine had to be told not to chase each other across me in their nighttime games, but I'm fine with their cuddling, wandering in and out, and occasionally walking across me or lying on top of me. (We compromised on their kneeding my belly -- the bigger one in particular tends to lean hard on sensitive spots, but I find the fact that they want to do so cute and flattering so I usually put up with it.)
5) The human will sometimes handle your toes, examine your teeth and ears, turn you upside down, toss you in the air and catch you, or otherwise do something odd. He's weird but harmless; don't panic. (I deliberately get them used to some of this under the guise of play so they're easier to handle when I need to trim claws or do something of that sort.)
6) Most humans are harmless. Some give good backrubs and/or give treats.
7) Humans are thin-skinned; try to avoid scratching them. Cats instinctively use their claws the way we use our fingertips. They need to learn when doing so is unnecessary or should be avoided -- when sitting on laps, for example, they should trust us enough not to have to hang on. I've found that gently pulling their paw-tips back is a pretty good "please don't poke" reminder.
There are other details, but they tend to be more specific to particular households. Mine understand "excuse me" as equivalent to"you're in the way; move". We've reached fairly specific agreements about my desk. And so on.
(Re the nipping question in the comments: it depends. I've known adult cats who like to nibble fingertips and do so gently; the only hazard is that an unwarned human may startle, startling them in turn and causing a harder bite than expected. So, like the claws, this can be negotiated, but you need to give the cat very clear signals about what is an is not acceptable, which means drawing a line with some buffer space around it, and recognizing that some cats need stricter controls than others.)
