There is an ocean of Russian reading material on the Internet. 
Before you plunge into it, though, try some simple reading exercises, described and provided on sites below. 

Cyrillic Alphabet - From the start: Russian alphabet and how to pronounce it with a sound file for each letter! 

The Russian Alphabet - Another site teaching the Russian alphabet. No sound, but explicit instructions on how to pronounce it. Especially appealing site with a list of very useful language links. 
 
The Russian Alphabet - A learning puzzle: challenging way to master Russian alphabet. 

One Hundred Russian Signs - "Eyes-on" reading. Quickest way to learn the language: READ the signs! How else will you know were you are going? Browse through Tom Beyer's slide show of Russian signs with English translations.   It's a great way to practice the Cyrillic alphabet while familiarizing yourself with the culture. 

Learn to Read Russian! - A reading and pronunciation lesson from Auburn University, accompanied by some sound files. 

Let's read Russian! - Irene Thompson from University of Hawaii publishes her textbook of sample texts and helpful strategies for learning to read Russian. 
 
Reading Textbook - Short texts from classic Russian literature followed by 
discussions. Excellent reading material for beginners. 

Russian Insta-Class - Electronic files of original brief news broadcasts in 
Russian followed by English translations with glossary of terms from SCOLA 
INSTA CLASS service. 
 

Animals in Russia - A Russian server dedicated entirely to animals - most 
complete with pictures and articles. Another great source for animal 
vocabulary. 
 

Please submit all questions and comments to Alina Makin