Although greyhounds generally have what is considered a "soft" temperament (ie, they are usually submissive towards humans, will not attempt to control situations and are very compliant), they are also hounds and because of this, they are often quite independent thinkers (a trait necessary for a dog that moves out of command range from its handler in work) and will not obey commands that do not make immediate sense to them.
Additionally, greyhounds are quite sensitive to correction so this combination can making training a pet greyhound challenging.
Although certain behaviours may be difficult to teach your greyhound (such as sitting on command), many of the basics needed to ensure a well-mannered dog are very simple and most greyhounds accept them readily.
Because of their soft natures, greyhounds learn best through positive training and being a fairly food motivated breed, this is quite easy to do (especially with methods such as clicker training, more information on clicker training can be found here).
Ideally, your greyhound should learn to understand:
Wait - A command given when the dog's food is put down, this command also helps your dog to learn the difference between things it can have and the things it can't.
Bed/on your bed - A greyhound that can be moved around without trouble is going to be a lot easier to manage around visitors, small children and in situations where you need the dog to move to a given position and you cannot physically move the dog yourself. It can also be used as a starting position for teaching a sit or stay.
Heel - For many dogs, the difference between whether they are walked or spend their lives in the backyard comes down to their ability to walk nicely on lead. Heel is an especially valuable skill to learn and will help with learning other positive behaviours also.
A lot of greyhound behaviour is either instinctive or habit, learned from the kennel environment.
Although they spend a large portion of their lives living with trainers and being "trained", this training rarely includes basic obedience and manners (sort of like the difference between an athlete training to run and someone being trained for a job; one is physical, the other is mental) so greyhounds straight from racing may be easier in some respects (such as house training and behaviour with other dogs) while being more difficult in other respects (for example, stealing food).
When looking at behaviour in greyhounds, it's useful to look at the behaviour in the context of kennel environment- a greyhounds that eats everything left out if not necessarily a badly behaved greyhound, it's just doing what it has always known to be right- in a kennel situation, the only food the greyhound will ever encounter will be its own, so, to a greyhound, all food left sitting around is intended for him.
When managing behavioural problems, it's also worth bearing in mind that while greyhounds may at times not seem very bright (they're admittedly prone to falling over and running into things, despite their elegant and graceful reputation), greyhounds are actually very intelligent dogs and will learn things that you might not necessarily want them to learn. This means that consistancy is very important and rules need to be enforced by all members of the family- an exception made here or there by a certain family member is usually not forgotten and will be gently pushed for again until the greyhound has effectively trained the human to comply (which is not an ideal situation).
When you adopt your greyhound, you will be provided with a summary of the dog's behaviour, outlining any areas that we feel may need continuing work (although these are usually quite minimal) and management methods will be discussed to ensure the dog has the best possible chance of learning to be a great member of the family.