As long as you practice common sense hygiene (washing hands after playing with puppy, picking up poo, etc.), you should be fine. Every puppy in the world has roundworms, regardless of where they come from - they pick them up from the mother when they nurse. Most treatments have to be given at least twice to get rid of all the worms, so make sure you follow up with your veterinarian and have a stool sample checked at every visit when he comes in for vaccines. The stool has to examined under a microscope for worm eggs, so you won't necessarily see anything yourself, but sometimes if they have large numbers of worms, they can pass some adults in the stool or even vomit them up (they look like strands of spaghetti). Sorry if that grosses you out! Just be sure your kids are washing their hands after playing with puppy and before eating because they could just as easily get exposed playing in the park and picking something up in the soil from another dog who had passed stool there at some point. Kids can pick them up and they can cause serious problems, including liver damage, and blindness if a worm happens to migrate into an eyeball. Again, sorry if this grosses you out, but also be assured that it is rare, and no reason to not own a puppy.
Also make sure you get puppy on monthly heartworm preventative - heartworms are a whole separate issue and can be deadly. The preventative medication that dogs are given every month should help prevent roundworm as well.
http://www.healthypet.com/PetCare/DogCare.aspx