Visas: Some visitors from Western Europe, Australasia and the USA can stay in Taiwan for 30 days without a visa. Everyone else needs a visa, which allows a 30-day stay. Single entry visas are easy to get, but if you want a multiple entry visa get it before you leave home. Because the Republic of China (Taiwan) is not recognized by most countries, you'll have to get your visa from a Taiwanese 'pseudo embassy' - look for trade offices, travel services or friendship associations.
Health risks: No particular risks, but you should consider vaccinations for hepatitis.