Greyhounds are very intelligent dogs that are bred to be quiet and graceful. These same traits that make them prized race dogs can also lead to some quirky personal traits. Many a greyhound owner has been surprised that their dogs appear to be sneaky when they first get them. This is due to the fact that the dogs are timid around strangers, when not fully socialized, and so might attempt to get comfortable in its new surroundings without getting permission first.
That means that when you leave any food out on the counter (even if the dog appears asleep) to do something and if you come back and the food is gone, the dog probably took it. It will even still appear to be sleeping at times even though the proof is that the food is gone and you were out of the room! They are this crafty and enjoy playing innocent.
Other quirky traits are collecting odds and ends from around the house. This may happen more with ex-race dogs than with a dog that you socialized as a pup. The race dogs have lived their entire lives without any personal items in the racing kennels and so treasure every new little item they can claim as their own. They won't understand that your kid's toys or the remote control are not to be hidden away somewhere, until they've been in your house long enough for you to establish boundaries. The safest thing to do is not to leave things lying around that will attract the dog to collect it.
Greyhounds will take every opportunity to chase anything that looks like prey. That means that if you haven't properly socialized your greyhound to your other small pets, you can expect it to chase them. Even if you get it to behave with your own pets, you should never leave the dog to run free in your neighborhood. It will chase the neighborhood pets and kill them sometimes. This isn't because it has an aggressive nature as much as a natural instinct as a prey dog.
A greyhound does need to be leashed when outside or at least kept behind a tall privacy fence. They may seem to be snoozing or paying attention to you one minute and the next they are off like a shot. They are easily startled by strangers too and can end up running away, rather than sticking around to defend themselves or you, so don't count on them as guard dogs. They can be socialized to accept strangers that you let into your home, but they will probably be a bit standoffish with them. They should also be supervised with your children when they are first in the home. Once they have accepted you as their family they can be very warm and loving, but it will take a period of adjustment to get to that stage.