I have found material that suits my level of Russian...
Gossip magazines.... ! Finally I have an excuse to read them.

I am also reading a parallel version of "Karlsson on the roof" (Russian - Swedish). It's really simple language and nothing fancy and poetic.. A bit boring though, I already know the story and it is for children.

At first I was thinking that I should throw myself onto the Russian classics or even high literature... That would be madness, I wouldn't understand anything, and it's not how people speak. It would be like starting ones English studies with reading Shakespeare and the King James Bible... a recipe for failure...

In fact, some of the first books I can recall reading (voluntarily) in English, were the super trashy "Flowers in the Attic" series, which women will remember, and Harlequin romance books.... Anything about Sci-fi or fantasy is not a good idea - it has to have everyday words in it, no specialist jargon.

Very simple and interesting enough that you will keep reading to find out how it ends.
I am looking for such a book in Russian, in fact.