Hi Cassie,
Thank you for posting your query in Microsoft forums.
From the description, I understand that you accidently unpinned Internet Explorer Tile and Weather App Tile from the start screen.
I would suggest you to follow these steps to pin the Internet Explorer Tile and weather App to the start screen.
a) Press windows + Q key
b) Type Internet Explorer
c) Right click on Internet Explorer and click on Pin to start option from the menu below.
Follow the same steps for Weather App as well. In the search box, Type Weather, instead of Internet Explorer.
Also Refer:
Pin or unpin apps
http://windows.microsoft.com/en-US/windows-8/shared-start-screen#1TC=t1
The same steps can be followed to pin the items to Taskbar if you are referring to pinning the items to Taskbar.
Hope this information will help you.
For any other Windows related issue, reply to us and we will be happy to assist you.