I have purchased a legal copy of Windows 7 Home Premium. I have all necessary codes (Key code, COA code, etc) How do I install it on a computer with no CD drive? I tried putting it on a memory stick, but it would not fit. Thank you in advance!

Since you didn't state the size of your USB stick, I would say that it is too small for the image. You need at least a 4GB USB drive. You still need access to a CD-Drive to properly create a bootable USB flash image. Microsoft has a page dedicated on how to create a bootable USB drive: http://technet.microsoft.com/en-us/magazine/dd535816.aspx