All the points raised by DBF would apply to any dog but if you are adopting a sighthound such as a grey then there are extra considerations, especially if you are taking one straight off the track.
We have three greys and I do a lot of voluntary work here in Wales. Some straight talking is needed especially on the subject of cats, guinea pigs and fences.
Your fences need to be able to withstand a grey jumping at speed - so solid, and at least 6ft tall. Any rescue that doesn't pick up on this at homecheck should be avoided like the plague. Gappy hedges, low wire fences, or low brick walls are just not enough.
Greys are bred and trained to chase after small, furry things. They do not distinguish between hares (artificial or real) and guinea pigs, cats or Yorkshire terriers. Please pause and think this through. It is possible to get cat friendly greys but they are much rarer - about 20% of them are potentially cat trainable. If you want your grey to co-exist with your GPs, I'd be thinking about how you can safely keep them seperate. A reputable rescue would be very careful about ensuring you were only matched with a thoroughly cat tested (and small furry friendly) grey and even then you should always exercise caution.
I am also going to be the voice of doom on the subject of cats. How will you feel if your grey chases and kills a cat that gets into your garden? It is impossible for you to totally cat proof your garden and it is also impossible to ensure your garden is cat free before you let the dog out for a wee. Many dogs like to chase cats but greys have the turn of speed to be able to actually catch them, and the wherewithal to be able to kill them. How will this affect your relationships with your neighbours? This is a common reason for greys to be rehomed particularly when people want to maintain good relationships with neighbours.
Sighthounds - we also need to discuss the dreaded "recall". Many greys can and do learn recall but this is not a natural strength and even when they do learn this, sighthound owners should ALWAYS be alert to their grey getting in the "zone" and hurtling off into the middle of next week. Once they start running, it is a breathtaking sight, but you will never catch them (40 mph) and they will be completely oblivious to anything round them such as traffic, cars or any other obstacle. For their own safety you will always need to to do this thinking for them and ensure they are only off lead where it is safe to do so.
Many, many people will tell you at great length about all of the wonderful features of greyhounds ( I often do myself) but rescues do adopters no favours if they are not completely honest and realistic about some of the potential downsides of a greyhound, especially one with a high prey drive.
There will probably now be a tsunami of people posting who have adopted greys who have got perfect recall and have never chased anything, and share their homes with a posse of rabbits and Chihuahuas - but as someone on the inside of rescue I've seen too many distraught owners whose dogs have chased cats, escaped and in the worst case been run over after suddenly taking off. There is also the issue that I heard recently of the RSPCA prosecuting a greyhound owner whose dog killed a cat while out on the lead (cat jumped out of hedge in front of grey). Owner prosecuted under Animal Welfare Act.
So, your rescue should really cover all these points. If they can get you a cat friendly grey, then you should be fine, but even then, I'd still insist on making sure you have a damn good fence. Sorry to be the voice of doom, but it's better you think about all this up front.