Defeating the Giant Spider: Circle around the Giant Spider in the Dark Forest and cast Rictosempra to defeat him. He will not shoot any spider webs at you.
Nimbus 2000 Broomstick: Go to the Quidditch practice on the second day of school and complete the training with a B grade or better to get a Nimbus 2000 flying broomstick. You will now be able to fly around the school during the day and can only land on grass. You will not be able to fly inside the school.