Methods
There are several different downloading options available to you. The 3 most popular methods are BitTorrent, Direct Connect and One-Click Hosting; a guide detailing these various methods can be found here.
Privacy
A great little tool to help protect your downloading privacy, especially with BitTorrent, is PeerGuardian 2. Available for MacOS X and Linux, as well as Windows, this software helps block any connections to organisations that might want to spy on your downloading habits. While this doesn't completely guarantee your anonymity, it certainly helps. You can download the software here.
Another line of defence is use something called a Virtual Private Network (VPN). Basically, a VPN routes your downloads through another network which hides your IP address, which is a very effective way of hiding your tracks. For more details on exactly how this works, check out the Wikipedia article here. There are many different services available online but one of the free ones is called ItsHidden. The only disadvantage of VPNs is that it usually will make your donwloading or uploading slower than usual. You have to weigh up the cost of anonymity against bandwidth. However, with some paid services, the speeds are so high, this shouldn't be a problem.
Another method for keeping your BitTorrent activities hidden is to encrypt your BitTorrent traffic. A great article explaining how you can do this for the most popular BitTorrent clients can be found here.
One final method is to use BitBlinder. This works by passing your request for data encrypted through multiple peers whereby each peer in the chain only knows the IP address of the next person in the chain instead of the original downloader or uploader.
So instead of this:

you have this:

You can download this software here.
ISPs
Something else that is very important to consider is your Internet Service Provider (ISP). If you are living on campus, this should not be an issue but off-campus, you need to look very carefully at ISPs if you are planning on downloading. As well as checking the obvious things, such as connection speeds, you also need to check out each provider's Fair Usage Policy (FUP). Some providers claim to offer you unlimited downloads when in actual fact, this is not true and you are limited by how much you can download each month. Other providers don't let you download large quantities during peak times. This is something you should definitely watch out for and consider so that you can find the right deal for you.
That concludes this guide. I hope you find the above information of some use.