If I'm downloading with BitTorrent and then shut down my computer midway through the download, do I have to start from scratch when I boot up again or does it just continue from where it left off?
No. The part you have downloaded will stay on your HD.
Just be sure to restart the program when you reboot. Most will auto start and auto resume on reboot. Check the preferences of each program. I use BitLord.
You've gotta be careful. Most internet providers have clauses whereby they can suspend/terminate your account if you're downloading too much. They target something like the top 5% of bandwidth hogs who use up 95% of the resources.