Listen to some Belarusian being spoken. It may seem counter-intuitive to start learning an alphabet by listening to someone speak, but by listening to the sound of a language, you will be able to better identify the sound of different characters when written with Latin characters. Try an online Belarusian radio service such as Svaboda.org.
Try learning the alphabet in Latin characters first. If you are a native English speaker, it will help to start with a familiar alphabet. Websites such as MyLanguages.org have a detailed alphabet chart accompanied by a pronunciation guide.
Familiarize yourself with the Cyrillic alphabet using a website such as Friends-Partners.org. This website has upper and lower class letters along with a pronunciation key. You can even click on the characters to hear them pronounced properly.
Compare the Latin alphabet to the Cyrillic alphabet once you are familiar with Cyrillic characters. Websites such as Omniglot.com have both alphabets, allowing users to learn through comparison.
Try obtaining a Belarusian language book. A Belarus Miscellany (belarus-misc.org) has a list of a few books that can help you learn how to read and write the Belarusian alphabet.
Try a language exchange to reinforce and check what you've learned. Speaking or writing to someone is the best way to practice what you've learned. MyLanguageExchange.com allows users to share any language through voice or text chat.