My goodness Ethel, I never realised you had such a violent streak in you! I'll have to be nicer to you in future :-)
JInglepot, the banking stuff is usually done through a secure site so there should be no personal banking info on your hard drive. Anyway, if you want to play safe you can install the old drive in the new comp providing there are compatible cable connection - IDE/sata. Installation is easy as even I can do it. You can then wipe the old drive with your disk management on the new comp and use the spare drive to backup vital info and personal piccies you never want to loose. You need to disable the spare drive when not in use and turn the modem off when in use so that you can't catch a virus on it.
It would be a lot easier to use an external HDD enclosure though, as Ethel said.
The parts of the old comp that you don't want can be taken to a recycle center, maybe one that rebuilds them for community projects.