Reading order

[Stephen]

Not sure if there is a correct order as such. The two works by Tolstoy and Doestoyevsky are pretty hefty, so if trying any of those first, you need to persevere and put some graft in. Have you read many other classics?

[Nick]

This is a good question, and I think the best answer is: there is no order! Classic Russian literature is often difficult and requires time and strength of will for many new readers. The best advice one can give, then, is to read whichever book seems most interesting to you.
That being said, I will always recommend Crime and Punishment as the first full novel of Dostoevsky's to read, as it becomes very exciting almost instantly, covers many themes that the modern reader is already familiar with, and is of a reasonable length.
Of your list War and Peace is my personal favorite, with the Karamazov boys coming in at a very close second. Both of these books will change your life, if you let them, but they do not go down easy. For War and Peace I read many articles and books about the history of the Napoleonic War era to give me a foundation from which to understand the Russia of that time. It was daunting, but made the experience much more enjoyable.

If you want to read the list you gave in a sort of chronological order (not by publish date by but by the actual events contained therein) this would be the read order in my best estimation.
1: Eugene Onegin
2: War and Peace
3: Fathers and Sons
4: Anna Karenina
5: Crime and Punishment
6: The Brothers K

If I may make a suggestion to any new reader: please, please read A Hero of our Time, by Mikhail Lermontov. Its one of the single most important, fascinating, and exciting, works from this era and every Russian author after Pushkin (so everyone else on your list) was influenced heavily by his tiny corpus of works (he died in a duel at 26). A Hero of our Time opens the door to the many nihilistic and 'superfluous' men you will see in Fathers and Sons and the Brother's K as well as opening up the world of the Russian Caucasus to you. He is painfully overlooked by Westerners who get into Russian lit, and shouldn't be. The book has the added benefit of being short, as well!

One final point: Eugene Onegin is a novel in verse, meaning that it is not so stirring in translation. To be frank, as a new reader I would not start with Pushkin, unless it was to read some of his short stories and novellas. He is the father of Russian Literature, and to the Russian people he is their 'everything,' but Dostoevsky and Tolstoy would be more deserving of your time at this stage. One should read Pushkin, indeed, one should read ALL of Pushkin, but only after an appreciation for the other great Russian authors has been instilled in you.
I would highly recommend watching Tchaikovsky's opera by the same title if you're into that sort of thing. You can find a truly superb version by the Met on Amazon.

Remember, there is no wrong way to start out on this journey, so follow your instincts.
These are my two cents, at least!