Lurchers are generally working crosses - bred for things like general pest control, lamping, hare/rabbit coursing. They're one of the most common type of dog you'll find in rescues, because sadly they get dumped when they are no longer of working use to their owners..
Because so many crosses can technically create a Lurcher (Greyhound x Border Collie, Whippet x Jack Russell, Greyhound x Staffordshire Bull Terrier) it can be a bit of a mixed bag, especially as most rescues can't really guarantee what breeds are in each Lurcher as they wouldn't know the history. Lurchers might be used for amateur racing, but as each one is unique there's no guarantees. I'd think with Lurchers, each one is an individual and would be fully temperament tested by the rescue to make sure they're a good match for your family.
Whippets are less commonly used than Greyhounds for racing, and are a bit harder to find in rescues. A Whippet is basically a medium-sized version of the Greyhound.
Greyhounds actually don't need a lot of exercise as they were built for speed, not endurance. They will zoom around for a few minutes but don't need extremely long walks. Some Greyhounds can be let off lead with a solid recall but others absolutely cannot, a lot of them have very strong prey drives and will take off after wildlife. Not all will, of course, but it's something to keep in mind. A long line or a safe enclosed area where they can run around without escaping is essential for Greyhounds that do bolt. In the first few months, I'd keep them on lead or have a long line (usually about 10-20m long) attached to the dog so that they have more freedom to run, but you'll be holding the end so that they can't run off. Unfortunately they are a breed where there is a real risk of you not being able to catch them once they start running! It might be worth signing up to some dog training classes, they will show you how to teach recall to your dog along with other useful things.
As all rescues come from different circumstances, some might bond straightaway, others may take longer. Some rescue dogs may have come from a loving family home due to personal circumstances and have been housetrained, some are ex racers who have only ever known living in a kennel, some may have been rescued from abuse (but I don't think a rescue would place an abuse case with someone who is completely new to dogs so don't worry). A lot of rescues will not be trained, so basic training is still essential. A good rescue will always been on hand with advice and guidance.
This is probably a bit too much of information overload, but most of it you learn along the way. I only got my first ever dog 4 years ago!