If the worms are alive then he needs worming asap. Use the wormer your vet gave you, which will be suitable for kittens.
The worms you describe don't sound like tapeworm, as these are segmented and look like little rice grains, so they're probably roundworms. These arrive in the kitten from their mother so are very common. The diarrhea is very likely to be linked to the presence of these parasites, you may find she also vomits them up. Not nice but her little body is rejecting them.
If you use Google Images, just google the term "cat worms" and you'll see the pictures of what you describe.
There's a great section in our Health and Behaviour module about worms, and how to prevent and treat them, but I'm very glad you've already seen your vet and he has prepared you well with the treatments.
As for the future, there are various flea and worm programs available so research products like Program, Advocat, Advantage, Profender etc and see which ones will suit your lifestyle and then check out the various on-line prices at VetUk or Animed Direct for good reliable supplies.
Do bear in mind that the dose will change as your kitten gets bigger, as all these medicines are dosed according to the animals weight and are easy to over/under dose if you get it wrong.
