This is a great question because the counter question is whether or not the cats actually train us. As kittens, you can train cats how to use a litter box by priming the box and helping them to dig in it after they use the bathroom. You can train cats about what time of day the food dish will be filled with yummy meat-based tasties. With a lot of work, you can train cats not to be nocturnal.
This is done by keeping your cat in a small room (bathroom) with the lights off and no toys at night, and then beginning play during the day. Especially if you start with kittens, it’s possible to teach them where it is acceptable to scratch – using catnip and helping them with the paw motion. There are some questionable training techniques about keeping cats off of counters using anything from sticky tape, to scat mats, to water bottles. In my house, the counter is actually a safe place so the cats can get above dog level. I have seen very calm and patient cats trained to walk on a harness and leash. I even know one awesome therapy cat who rides in a stroller and goes to nursing homes. But seriously, cats are very smart; they are also very independent.