Suggestions, PLEASE!

If any of you other mothers out there can give me any suggestion, I need them. Riley Rose has started having 'temper tantrums'. Already?!?! I know. She's only 15 months old! So....this is her pattern: She wants something, she screams the highest pitched scream you can imagine. The scream lasts as long as it takes you to get whatever she wants to her hands. If it is something she cannot have, for instance, she points at the kitchen cabinet because she knows the Cheese Puffs are in there (her new favorite food) she screams that horrible scream even when we tell her "no Riley Rose". This morning we had an incident where she wanted the toothpaste. She has always loved to chew on a toothbrush so we bought her one that is for babies. That was fine, but now that she knows that we put that 'stuff' on our toothbrush, she wants it too. She knows how to open the tube and she SQUEEZES it everywhere. She screamed for, I know, 7-8 minutes becuase I took the tube of toothpaste away from her. The only reason she stopped was because we walked outside to leave the house for school/daycare.
Poor Kaylee too. I feel so bad because if Riley Rose wants something that Kaylee has (which is often the case), I catch myself saying "Kaylee just give it to her so she doesn't scream". I know this isn't fair to Kaylee either.
Randy and I have tried everything we know to do. We have talked softly when she screams and told her to "sshh", we have firmly pointed at her and said "NO", we have tried ignoring the screams (which was the most difficult), and now we have started a time-out chair which she sits in for 1-2 minutes during these episodes. Nothing seems to work! We keep telling ourselves that she will outgrow this soon, but how soon?
Any suggestions????