Puppy worms

Okay, new situation.
I took Phoebe to the vet yesterday for her first shots, check up, and worming. He told me to keep an eye out for little segments of tape worm. Haven't seen that but lots of long spagetti type worms. Yucky to be sure. My question is this. He didn't say anything about these type of worms. Are they heart worms? Of course its the week end and I can't get hold of him. She seems to be fine and I am glad that she is eliminating these ugly things but I am concerned now if I should be taking her back in for some additional treatment. Her next appointment is in a month for more shots and probably another worming too. I am just ammazed that she has these horrible things inside her.
Suggestions please.
Thanks you all.
Jan