This will not, perhaps, be very helpful to you, future reader, to hear but: in my humble opinion, there is no way to *learn* to like Tolstoy. There's no process of adjustment, no method of accustoming oneself to the prose, the descriptions, the style, the themes. It's either there within you or it's not.
In other words, if you begin "Anna Karenina" and you are not immediately swept up into the story, with its many characters, family tensions, and ornate depiction of Russian society on many levels... If you are ten chapters in and going forward on pure stubbornness... Put the book down. Walk away. This is not for you.
For example: I read in an earlier review that the reader was "bored" by Levin's description of working in the fields with the peasants on his estate. Personally, I find that to be one of the most compelling passages in the entire book. I'm not right while the other reader is wrong, but I will say this: it's a matter of taste. If you are not engrossed by the complexities of this vast and entrenched society, if you do not feel sympathy for Levin, or feel drawn to Anna, or understand the attraction of Vronsky, then do not torture yourself, and move on.
If you're staying, though -- Anna remains, I believe, one of the most interesting protagonists in literature, and precisely because while the reader is almost unwillingly forced to sympathize with her feelings, it is similarly impossible to remove the stigma of blame from her, watching the wreck she makes of her life. Her transformation from the alluring and enchanting woman who so impresses young Kitty, to the sad and scorned woman that Vronsky himself no longer truly loves, in the end, is all of her own doing -- but who among us can say we would have successfully avoided all of her misjudgments?
Contrasted with Anna is Levin, though their lives are intertwined only through friends and relatives and they have no real knowledge of each other -- Levin is Anna's exact opposite. We meet him as an awkward and abrupt, solitary man, with troubled family relations and an unrequited love -- and in the end, after his long journey of self-awareness, we leave him in a place of pure contentment. We warm to Levin and take him to our hearts, perhaps because his choices are the ones we would *like* to think we would make.
If you ask the average American to name a Tolstoy novel, they will generally say "War and Peace", but I've always thought "Anne Karenina" to be the more human story, the more accessible, and perhaps the greater classic because of that. It truly is a matter of taste -- but if it's to yours, you'll have stumbled upon a literary find you'll treasure always.