Are Roundworms contagious?

No they are not infectious, they are not a disease. The eggs can be ingested and thus travel to the gut where they live and reproduce.

The eggs are expelled in the faeces of affected animals and remain viable on the ground and in grass and vegetation for a significant amount of time.

Puppies can be born with worms if there mother was affected. Puppies can pick up worms if they live in an environment where affected dogs are allowed to go poop. They step on the faeces and then lick their paws.

You should always worm a puppy at the same time as you get its first vaccinations, then follow the advice of your vet on followup treatments.

Tapeworms are slightly different in that they are expelled from the rectum, not necessarily in the faeces, so the expelled segments can be ingested by other animals, pretty much anywhere.