The Cyrillic alphabet is usually the easiest alphabet for foreigners to learn, due to its similarity to the Latin alphabet. Cyrillic is the script used by many languages, including Russian, Belorussian, Moldovian, Serbian, Ukrainian, Kyrgyz, Kazakh, and Mongolian. It was developed in the 9th century by Saint Cyril, derived from the Greek alphabet, the same alphabet the Latin alphabet was derived from.
The best way to learn Cyrillic is to make yourself familiar with the letters, know which ones are the same, which ones are switched, and the new ones. Once you've got that down, start by just writing words in English but in Cyrillic. I learned it in High School, so I would practice by passing notes written in Cyrillic English to Russian friends, which actually works quite well, for you underageb&'s out there. Make sure you're writing it down phonetically, not literally. Once you've got writing them down, so you know the sounds off the top of your head, just start reading Russian news sites. You will be incredibly surprised at all the cognates in Russian, hidden in their Cyrillic mask. In 2010, about 40% of Russian news articles were cognates. You can download a keyboard map for Windows that is phonetic in various places on the internet.
Гоод лук, комрадс!