Time Machine backup disks should be external ... separate from the computer.
Several places sell external disks that are already formatted for Mac. Others tell you that external disks not formatted for Mac can be reforatted, but save yourself the grief for now and buy Mac-ready.
Buy a *new* disk, in my opinion. Do not reuse an old disk.
Some can suggest certain brand names as "better", but even the best disk-brand will have bad apples in the mix.
If any others have a convenient link for Time Machine Tutorial, please provide. I have learned enough alreay, generally through trial and error on my own.
But focus on getting that first backup done within the next 2 days.
As to "probably need a new HD", perhaps you can move enough off to an external disk. Cross that bridge after you get the backup.
If you get the sense that critical things are done first and that you should save yourself complex things like "how to replace a harddisk" for later, you are reading my posts correctly.