If you start with an all-new PC complete with operating system, you can connect your old drive as a "slave" (pah! to PCness) drive D, and read stuff off it.
Personally, I wouldn't try to start a new PC with an old hard drive. That way lie many hours of chasing bugs, and never really finding all of them.
If you still have the install disks (or activation codes) for the software you have installed on the ailing machine, you can reinstall on the new one.