You must know the sounds (regular plus exceptions) of Russian letters to be able to read the language.
I think it is best to learn to pronounce Russian words properly before trying to read them, because there are exceptions to the simple reproduction of the alphabetic sounds. Reading them first, using the sounds from the recitation of the alphabet will introduce error. Native speakers learn the language this way, first by speaking it with their family and then to read it from their teachers.
https://en.wikipedia.org/wiki/Russian_orthography
You can learn to recite the alphabet and associate the letters with the sounds by singing an alphabet song while keeping track of the letters.
https://www.google.com/webhp?sourceid=chrome-instant&ion=1&espv=2&ie=UTF-8#q=Russian+Alphabet%2C+Sesame+street+song
It is best to write Russian using the standard Russian keyboard layout. You may remap your keyboard to do that, There are other methods that allow you to select letters with a mouse or even to hand write using a stylus. .
http://learnrussian.rt.com/speak-russian/how-install-cyrillic-keyboard-windows-7
The big exceptions in writing Russian are the unvoicing of some consonants at the end of words, the replacement of the sound of letter "г" with a "в" sound in the pronoun, "his," "его" and in the ending of the masculine animate genitive or accusative "его." Vowels are also reduced from their value from reciting the alphabet depending on stress. o > a > ə.
The Russian word for "good," "хорошо" shows this. Russian has the sound of the schwa, but only occurs when another vowel is reduced, so Russians don't write it. Written phonetically, хорошо would be written хəрaшо.
Despite these exceptions, Russian spelling is far more phonetic than English. It may just be my opinion, but I think it is better to learn how to actually pronounce the words and then how they are written rather than learn all these rules. The alphabet is a very good way to remember the pronunciation, but not definitive.