It's not important what port you use, so long as the port isn't used by any other service. BitTorrent is supposed to use a randomly selected port to get around any port based blocking.
What is important, for best results, is that whatever port you do use is forwarded by your router. There are various online guides to port forwarding if you're unsure about this.
Using a good BitTorrent client (uTorrent is one of the best) can help to some extent.
However, ultimately, your download speed can only ever be as fast as the combined upload speed you can get from the people who are seeding the torrent. That's why seeding is so important, and why the good torrent sites hate leeching so much.