I'd highly recommend having a read through the various puppy related resources on the Doghouse sticky thread, I think they might be quite useful to you.
Puppies are a lot of work. House training is very involved early on, you can be taking them out every few minutes trying to catch a wee out in the garden sometimes. Most are very bitey, it's a normal and necessary part of their development but can be very hard to deal with sometimes. They'll destroy things left in their reach, anything remotely interesting is fair game and they can turn a book into confetti remarkably quickly. One of my three whippets was terrible for chewing the furniture as a puppy, another in a quiet moment managed to pull up a thread on the carpet and unravel a big patch of it. 'Puppy zoomies' also takes on a whole new meaning with a sighthound puppy, think wall of death round the room interspersed with mountain goat-like scaling of the furniture... If you're not phased by the idea dealing with a puppy plus small children then go for it, but definitely go into it with your eyes open. Hope for the best, prepare for the worst as they say!
As for whippets and cats, lots of people do keep them together but you do need to be realistic. Sighthounds and cats are always going to require some level of management and the potential is there for ending up with a dog that needs to be kept entirely separate from the cats. It's also worth noting that cat safe in the house doesn't necessarily equal cat safe in the garden, even around the family cats, and rarely translates to them being remotely trustworthy around strange cats. This is a good little article on keeping high prey drive dogs with cats, it mentions terrier specifically but it's all applicable to whippets as well. The temperament of the cats will also have an impact on how successful integrating a sighthound into the family is likely to be. It's a much easier (and safer) process with very confident, dog savvy cats than more skittish ones who might be prone to bolting.
Oh, and if you go for a whippet teach your kids to stand still when the dog is really running. Trust me on this, they mostly know where the things are they need to avoid and trying to dodge out of their way is more likely to result in a collision. I'm not small (5'10" and play rugby league at prop forward) and I've been totally wiped out by a dinky little 12kg whippet going at full speed into my legs because I tried to get out of their way 