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You want to get uTorrent or Azureus(vuze), uTorrent is a more basic client but will do everything you want it to and uses less resources whereas Azureus uses java and hogs your system a lot more but has the potential of more tinkering(pretty much unnesscary)
As far as downloading goes your much better of registering at somewhere like bitsoup or ilovetorrents ( google them ), while they arnt private sites as pretty much anyone can join, they have user limits and the rips of whatever material are scene releases (groups of people who ensure quality of the product) and there is a better seeder to leecher ratio meaning you get your files faster. Piratebay and other public trackers like isohunt etc are full of dodgy rips that are often mislabled and the people seeding only want to give you 5kbs and share with a million downloads = bad speeds. For the sake of 5mins signing up you can get your stuff quicker and easier. However you will be required to seed the files on these other sites but its pretty easy just dont go crazy with downloading at first. I manage to maintain 2.0+ ratios on these sites with barely doing anything. Even better speak to your friends and see if you can get onto a higer level site and get some advice on how to set up things better.
Also if you have the time and know how follow the tutorials on portforward.com to open up a port on your router that you can use for bittorrent. This will vastly improve your speed as you will be able to connect to everyone as opposed to just other people who have chosen to open their ports.
Hope that helps. peace