We can't guarantee a house trained dog but most Greyhounds are clean in their kennels, and once they know where you want them to empty, they will be happy to stick with that routine.
House training should begin as soon as your dog arrives home. Take them straight into the garden, wait until they relieve themselves and then reward them with a small piece of cheese or a dog biscuit.
After this initial visit to the garden, keep repeating this routine at regular periods throughout the day. It is advantageous to take your new pet for a short 5 minute walk every hour on the first day, every hour and a half on the second and every 2 hours on the third. During this time it is unlikely that the Greyhound will have had a chance to have an accident and they will be thoroughly familiar with the idea that any 'toilets' are to be done outside.
Thereafter, if your dog has an accident indoors, bear in mind punishment does not work and can make the dog worse. Anticipate when your dog needs to go, take them outside and praise and reward them when they 'go'.
Any 'accident' in the home should be washed thoroughly with a solution of biological washing liquid, as this will take away the smell, otherwise they will constantly re-mark over that spot.
Take them to the toilet immediately after food, when they get up and before they go to bed and, of course, in between.
Some signs to look for when your dog needs to go to the toilet are restlessness, pacing up and down, whining, scratching at the door or circling.
Of course, at first, there may be no signs as the dog will have been used to living in a kennel, but Greyhounds are generally clean animals and learn very quickly.
Remember, all pet owners have a responsibility to pick up any mess your dog does while walking anywhere on public ground. It's not only illegal, with fines to be paid, but also very nasty to tread in and offensive in a modern society. Poop scoops can be bought from your local pet store or vet practice or you can use 'Nappy sacks' which are a cheaper alternative.