Localbitcoins.com – You can buy bitcoins from other individuals either in person or using bank transfer. There is an escrow system in place for protection of buyer and seller. Using bank transfer you dont even need to leave your house, you can do it via online banking.
BTC-e.com – This exchange does not require verification to deposit money or trade but you may need to provide verification before you withdraw. I sent pictures of my photo driving licence and a photo of a utility bill and was verified within 48 hours.
Bitstamp – you need to be be verified before you can deposit money. Follow the rules they lay down re: high quality scan and photo and you will be fine.
MtGox – Same as above, follow the rules and it is fine. Verification did at one time (April bubble) take a few weeks, it is much more reasonable now.
These are 3 of the exchanges but there are others such as Kraken, Coinbase etc.