Most cats require little to no litter training, as litter boxes are designed to appeal to their instinctive need to bury their waste. As long as the box is of a sufficient size (larger is better than smaller) and the litter selected doesn't bother them (declawed cats or cats with sensitivities may not like many types of litter), then there should be very little effort involved, if any.
If you're adopting from a reputable rescue or shelter, the cat should already be litter trained; the shelter and foster families aren't just letting them go anywhere they want. An adult cat will also have a set personality and behaviors, and is less likely than a kitten to develop litterbox issues after adoption. Adult cats often have a harder time finding placement than kittens, often because of a perception that they've been surrendered due to litterbox issues; a rescue will be able to help you find a cat with a suitable personality and no known issues.
When bringing a new cat home, a good way to help it acclimate is to restrict the space available at first, and to ensure the cat knows where the litterbox is on arrival in the strange new space (litter attractants can also help during this period). It's often recommended to set the cat into the litterbox when letting it out of the carrier on the first arrival. Let the cat have space and time in the new room; let it hide and don't harass it or try to force it out of hiding. Leave the room entirely so it can get comfortable, and it'll use the box on its own.
Additionally, be prepared to clean up any accidents appropriately during the settling period. Don't scold the cat for them, and do clean them up with an enzymatic cleaner designed for cleaning up pet waste, or that area will begin to smell like an appropriate toileting area to the cat.
You can also minimize the chance of future litter issues by making sure the cat has plenty of clean water available and ensuring they are well hydrated (feeding wet food is very helpful for this); most litter issues are actually related to health problems, not behavioral, and most of them originate from urine that is not sufficiently diluted. Promoting dilute urine helps prevent these health issues.