The first thing to remember is not to make a fuss of the dog when he/she first arrives in your home.
We know that this is hard and goes against what you may want to do, but you need the dog to find its own feet around the house and settle happily without becoming too dependant on yourself. So for the first few days you must be hands off, if you already have existing dogs let the new dog take its lead from them, after all you need them to slip into the pack and if you constantly reassure the new dog, then its going to be bonding with you first and foremost and not them.
They should see you as the provider, the one who feeds, cares and walks, the other dogs should be companions first and foremost not you. Once the dog has settled into the pack happily then you can start to indulge it a little bit more. Make sure the dog has a quiet place it can take itself off to over the first few days. A quiet corner with something to lie on away from the hustle and bustle of the other dogs, the children. Remember the dog is going to be confused at first and things will seem strange and overpowering. If the dog feels too much responsibility it will become overwhelmed and this could lead to behaviour traits.
The dog may choose not to take itself off and settle easily in the same room as you, but if it does want time out and you have provided the bolt hole, so to speak it, will lead to happier settling.
From the very first day the dog arrives with you make it have time out on its own, during the course of every day. Use the bolt hole tactic again and find one room with a smallish space, (e.g. kitchen), provide a bed and feed the dog there also.
Top tip is probably to not shut a door on the dog' Greyhounds can become stressed when presented with a solid door. These dogs have been kennelled during their racing lives but they can still see out. Shut doors can sometimes result in scratching, panic and damage. You can if you wish use a dog gate or a baby gate. Dog gates are higher than standard baby gates and you can also open them to walk through.