Our adoption process was really simple. We visited our local greyhound trust one Saturday and they'd selected four dogs for us to walk that afternoon. We liked one in particular, so they brought her to our house for a home visit the next weekend. This was partly for them to check our house (whether the garden was secure, etc) and for us to see how she responded to the house. Greyhounds have often never been in a house before, so ours was unfamiliar with stairs and surprised by noises like kettle, washing machine, tv, piano.
We really liked her and the charity were happy with us, so we arranged a date for her to come to us permanently. Around 10 days after she arrived, we confirmed with the charity that we were happy to have her and they made all the paperwork and microchip over to us.
The charity are really supportive and have given us advice on a few things like barking and how to get her to jump in the car. We go on the monthly greyhound charity walks with her to keep in touch with our local greyhound community.
In terms of spending time with her, she arrived in the middle of the Easter holidays so we were around a lot. We decided to get her in the spring because DH and I can often work from home in the summer term (we are academics) and so we have around 6 months of being around quite a bit. I also work part time.
How we are doing things at the moment is walking her for about 30-40 mins before breakfast, then we all go out to work/school. DH comes home and gives her a lunchtime walk around 12 or 1, then ds and I get home around 3:40 and walk her again. Or sometimes one of us works from home all day. She does sleep a lot, so has been ok being left for around 5-6 hours once or twice. Mostly it's around 4 hours, though we will probably increase this in autumn term as she's more settled. We have a webcam on her so can see that she sleeps when we aren't there. We leave classic fm on for her so she's not in silence.
She's settled really well, and understands what she can and can't do. She doesn't try to eat ds's lego, and is happy to lie around wherever we are.
Hope that helps, happy to answer other things?