Great classic writers usually have their own personal styles that require some time to adapt to and also usually have an overexpansive vocabulary, so no, pure classics of XIX century is normally not a good idea to start with (unless there are adapted versions for children or foreigners). Dostoyevsky is especially bad in the beginning - as a matter of fact he created his own language very different from either literary or spoken Russian of any age. Turgenev ("Diaries of a Sportsman (or Hunter)") has quite a standard language. Maybe that's why I don't like him . I found his writings boring and pointless but it is considered to be a part of "Great Classics" that many people enjoy.

I recommend to start with a children literature. Russian children literature has great traditions, many classic writers wrote tales for children specifically and it is often a part of "Great Classics".

Start from Lev Tolstoy's tales for children. It is good for the very beginner (that's what little children start to learn reading with), but even perfectly adult native speaker can enjoy clear language and narrative. Then there is a bunch of quality children literature from XX century starting from Alexey Tolstoy's Golden key. Also in USSR there was a really great school of fiction translators so usually you can read world literary classics, which you like, in Russian and be sure that it is of a quality and at the same time in a quite standard language without hardly comprehensible local color. Including again children literature: Winnie-the-Pooh, Tuve Jansson's tales etc..

Beside children's literature I can recommend Chekhov's short stories. They are quite clear and exact if you know what I mean. And short. Though somewhat bitter.

Also a recognized absurdist Kharms has surprizingly clear and exact language. And again very short pieces of text to enjoy.