I have a 1 TB hard drive; HP pavilion g7 laptop; OS windows 8; have not upgraded to 8.1 as of yet.

What would be some options for setting up a backup system where once a week or so I could connect to it? Thus if needed to restore any or all of the hard drive it would be relatively easy. Is Seagate recommended? Would it backup everything including the os files?

A system image backup would cover OS files. A standard backup would not in most cases. Your options for backing up would be:

a) Use the existing Windows Backup function
b) Buy backup software and/or drive imaging software
c) Upload and use all of your personal documents and files in the cloud and use Windows Restore Points
d) Clone your hard drive, once a week
e) Mantain your own FTP server, and backup there (Requires paid programs, as far as I know)
f) Use an existing online backup sytem (Requires a fee per a certain time period)