Getting into Serbia from everywhere but Kosovo is simple and should be hassle-free. It may not be possible to enter Serbia from Kosovo unless you first entered Kosovo from Serbia. Some travellers with Kosovo entry or exit stamps in their passports have been denied entry to Serbia; check with your embassy for up-to-date information.
None required for citizens of the EU, UK, Australia, New Zealand, Canada and the USA.
Tourist visas for stays of less than 90 days aren't required by citizens of EU countries, most other European countries, Australia, New Zealand, Canada and the USA. The Ministry of Foreign Affairs (www.mfa.gov.rs/en) has full details.
Officially, all visitors must register with the police. Hotels and hostels will do this for you but if you're camping or staying in a private home, you are expected to register within 24 hours of arrival. Unofficially? This is rarely enforced, but being unable to produce registration documents upon leaving Serbia could result in a fine.