"I can swim quickly." In some dialects people don't use adverbs, so you will hear people saying "fast" instead of "quickly". Please don't speak like that, it sounds horrible to most native speakers.

I think you could use either. I would suggest "I am a fast swimmer or I can swim quickly".

If you read sports news, you will see that most writers use "fast". Here is an interesting page about swimming fast: http://swimming.about.com/od/freeandback/a/free_swim_skill.htm

I think quickly is more suitable

Either is fine!