My experience is that Lurcher pups are mad as a box of frogs but also really loving and sweet and great fun. Of course a lot depends on the breed mix. Saluki crosses tend to be more sensitive, collie crosses brainier and capable of more mischief.
First bit of advice is to, buy Total Recall by Pippa Mattinson and train recall to a whistle. It’s much easier to get them to return to a conditioned response than your voice when they are full flight with the wind in their ears. I did this and mine had excellent recall.
Expect the ‘scream of death’ over tiny incidents but zero injury. They will make you panic, thinking surely they must have at least broken a leg, only to be absolutely fine two seconds later after a fuss and kiss better! 
. On the other hand, thin skin does equal potential for nicks and scrapes, so don’t be surprised if you have a few trips to the vets in the first year while they are acclimatising to their speed and getting past the gangly legged bambi stage..
You, your family and potentially friends, will probably get ‘Lurchered’ (feet knocked out from under you) at some point while pup is learning to control speed, turning and brakes. If they’re hell bent on zoomies around you in an open space, stand together as a group, as that makes you more visible at speed.
If you possibly can, hire secure fields where you can let them run safely. Obviously you have to be careful when they are still growing and developing, but a sighthound in full flight is a thing of beauty. They do need to run, it’s in their bones and makes them so happy. They don’t however need to run miles every day and you will probably be surprised how much they sleep.
Learn to put anything you don’t want snaffled properly out of reach. Worktop surfing is a Lurcher art form. Mine were caught standing in the middle of the dining table as well when they were pups.
Most Lurchers hate the cold and wet, so coats are a must. Those with very thin/short greyhoundy fur may appreciate a fleece at night as well.
Make sure you have an endless supply of fleece blankets. They love to stretch out so flat beds are better than those with raised sides. Fully expect them to forgo said bed for the whole of your sofa at every opportunity.
My Lurcher pups loved soft toys best, but be warned they will be destroyed relatively quickly. The ones you can buy without stuffing fare better and they tend to love anything with wings (ducks/pheasants etc) as they love the way they flap when they shake them.
Don’t be surprised if their playing style seems unusual and quite rough compared to other breeds. I remember that coming as quite a shock to me with my first. Mine played that way happily together (lots of fast chasing, plus tail and neck chomping) but wouldn’t have been popular if I had let them behave like that with the local Labradors and Spaniels etc.
I’m just off to bed now, but If I think of anything else I’ll post again tomorrow.