I'm not sure I can even discuss Daoism by itself briefly, even though I know almost nothing about it (and, I might add, I may know some things that ain't so).
I guess the too-long-didn't-read version of Daoism would be that in pure Daoism, people follow the Chinese philosopher Laozi and his work, the Daodejing. Laozi and the original Daoists sought to become detached from the tumults and troubles of the world and to live a life in harmony with nature. But I think that most modern Taiwanese seem to associate Daoism with traditional Chinese religious practices, an association that I imagine that some of the original Daoists would have found appalling. Traditional Chinese religion seems to me to be a colorful, noisy, exciting thing, whereas the original Daoism seems like a quiet, withdrawn sort of thing.
Daoism and taoism are two spellings of the same set of beliefs. The reason that it is spelled two different ways is that in the old days, when Chinese words were written in the Roman alphabet, a system called Wade-Giles was used. In that system, the sound similar to our t was written as t when it was breathless, and as t' when it was accompanied by the breath. But over the past half-century or thereabouts, a writing system called Hanyu Pinyin, or just pinyin, has come to be a very popular system for representing Chinese sounds in the Roman alphabet, and by that system, the t-resembling sound is represented by d.
What little I might know about Daoism comes from an old (1969), free book (that I never finished) written by a scholar named Wolfram Eberhard, and entitled A History of China. You can read what Dr. Eberhard says about Daoism on pages 45-50 of his book, which you can find here: http://www.gutenberg.org/ebooks/17695
There's also stuff about Daoism in Wikipedia:
Here's some stuff in Wikipedia on Wade-Giles and Hanyu Pinyin:
I guess the too-long-didn't-read version of Buddhism would be that Buddhism was founded by a man named Siddhartha, who I guess you could say was an Indian (even though many pilgrims visit a place in Nepal as Siddhartha's birthplace), and who reportedly lived in the sixth and fifth centuries BCE. Siddhartha seems to have become very unhappy because of all the suffering and futility in the world, and seems to have abandoned his position and wealth to pursue a new approach to life. Accounts of his life (again, about which I know almost nothing) seem to say that he attained some kind of transcendent state through meditation. At some point, he developed a belief system that involved, to put it in extremely simple terms, diminishing one's worldly desires, and he began teaching these beliefs to others. Siddhartha became known as Gautama Buddha, or just Buddha. In Chinese I think the easiest version of his name is Fó, or Fótuó.
According to Dr. Eberhard (on pages 124-125 of his book), initially traders from Turkestan and India made it possible for Buddhist monks to establish themselves in China.
I think that among a lot of Taiwanese, Buddhism and Daoism tend to get kind of get identified together, at least to some degree.
Here's some stuff on Buddhism in Wikipedia:
I hope this helps.