Yes, it sounds like roundworms. Roundworms typically average from 3-6 inches and resemble spaghetti. And yes, tapeworms are rice like segments that resemble seaseme seed when dry up.
I would recommend taking a stool sample to the vet and having it checked for any other internal parasites. Other parasites are not visible to the naked eye and will need to be checked for under a microscope. Different medications treat different combination of parasites, so you'll want to know for sure what needs to be treated.
Different worming medications have different instructions. Some are liquid and require several days of dosing, and some are tablets you give all at once. In most cases, worming medication will need to repeating in 2-3 weeks since the meds only have an effect on the adult worms. 2-3 weeks is enough time for the eggs and immature adults to develop and be killed with the second dosing. In severe cases, additional worming may be necessary.
Make sure you are scooping the litter box daily and preferably twice daily. Cats can easily re-infect themselves from the eggs in the stool and be sure to wash your hands frequently, because roundworms can infect you too. Some doctors will will have owners with multiple cats treat the entire kitty family since more often then not, they all use the same litter boxes. Just check with your veterinarian.
Also, make sure you monitor the water intake because any form of a loose stool can lead to dehydration problems if they persist. I hope I was able to help!