As a new greyhound owner
The most important thing to remember, and I can't stress this enough is that a dog will NEVER do anything without a reason.
A dog has a reason for doing EVERYTHING he does. If a dog does something a person believes was for no reason, then they are wrong - the dog had a reason, but person hasn't picked up on what it was.
The second important thing is that a dog is a pack animal and requires a leader to follow. You are now his pack. 99.9% of all dogs require a pack leader to feed them, give them security and guidance. If a dog believes it's owner isn't pack leader material (calm and assertive) then he will feel he has to do it himself. If ever you hear someone say 'my dog never listens to me' then the dog doesn't regard it's owner as pack leader.
Remember that some everyday things you or I take for granted may cause your new greyhound much anxiety. This is because it could be the first time they've seen or heard such a thing, or been in an environment like this before. (A similar scenario we could find ourselves in is going abroard for the first time, going into hospital for the first time and your first day at school).
Give them time to learn to understand what you want of them.
Although some may know their name they may not recognise it in your voice or accent. Give them time to learn the tones of your voice.
Talk to them because even if they don't understand your words or accent, they will be able to understand the tone of your voice. High is happy, Deep is unhappy.
Don't be angry with them, don't give them 'time out' and don't lock them up as a punishment - You have your family, friends and hobbies, but they only have YOU.
Achieve what you want of them using praise and reward for good behaviour.
It's crucial for their well-being that you learn to place your trust in them.
Remember that however you treat them they will never forget it.
Before you hit them remember they have teeth that can easily hurt you if they choose to bite you.
Before you scold them for being lazy or not doing what you ask of them, ask yourself if something might be bothering them such as being on the wrong food or having their bed in an area they feel unhappy with - remember a dog does nothing without a reason.
A Greyhound is the longest living of the larger breeds of dogs and can live to between 10 and 15 years, so any separation from you will be very painful for them.
Love them and take care of them as they grow old. Remember you too will be old one day and will want someone to take care of you.
If they have to go somewhere that is 'difficult' never say "I can't bear to watch" or get someone else to go in your place. Everything is easier for them if you are with the and please remember they love you for rescuing them from perhaps a hard life or even and early death and will forever love YOU for that.