You know how, when you're playing a video game, there's sometimes a demo section that lets you poke around the game's world and figure out how everything goes before you get on to the levels that really matter? Dolly's kind of like that in relation to the rest of Anna Karenina.
In the first sixty pages, Dolly has to deal with adultery (her husband Oblonsky's), forgiveness (as a foil to Karenin), and motherhood (as a foil to Anna). Most of the major themes of the novel come up in Dolly's story, but everything is on a small enough scale that it seems a lot easier to handle than the next 750-odd pages of the novel. Dolly's like our training wheels for reading this book: she gives us the time to figure out how Anna Karenina handles before we get thrown into the main body of the book.
Dolly's story intersects with the big Tragic Love at the heart of Anna Karenina three times. The first is in Part 1, when Anna comes to mediate between Dolly and Oblonsky. At this stage in the novel, Anna's a good woman (according to Tolstoy), so she's primarily concerned with continuing and strengthening the family unit. And it's her advice to Dolly to forgive Oblonsky that makes Dolly so loyal to Anna later in the novel. Dolly is essentially the last woman in the book who's willing to talk to Anna.
You know the saying "One good deed deserves another?" Well, Dolly's next intersection with Anna's Tragic Love Story comes up as an example of this saying. Having had her family life saved by Anna, in Part 4, Dolly meets up with Karenin and attempts to return the favor. She pleads with Karenin to forgive Anna and keep their family together. She says:
"No, wait! You mustn't ruin her. Wait, I'll tell you about myself. I was married, and my husband deceived me. Angry, jealous, I wanted to abandon everything, I myself wanted ... But I came to my senses – and who saved me? Anna saved me. And so I live [...] I forgave, and you must forgive!" (4.12.38)
Here, Dolly begs Karenin not give up on his marriage. He should turn the other cheek and not give in to his anger and jealousy. Dolly's conversation with Karenin at this critical juncture before he goes to see Anna during her illness may be one of the factors contributing to his own (brief) religious illumination.
And then comes Dolly's final exchange with Anna. In Part 6, Dolly is visiting Levin and Kitty and realizes that their estate isn't so far away from Vronsky's. At this point, Dolly, like Karenin, has kind of gone back on her forgiveness of her cheating spouse, and wishes that she'd tossed Stiva out on his ear instead of taking him back in Part 1. Dolly thinks sympathetically of Anna:
And they all fall upon Anna. What for? am I any better? I at least have a husband I love. Not as I'd have wanted to love, but I do love him, and Anna did not love hers. How is she to blame, then? She wants to live. God has put that into our souls. (6.16.17)
There's so much going on here: first, Dolly is shocked at the way Petersburg society has treated Anna. And Dolly's thinking that her own love for Stiva is as a result of their circumstances. She's beginning to realize that it's because they're married and he's the father of her six children that she loves him, and that she doesn't love him in a "real way." Furthermore, Dolly can understand why boredom drives Anna to "want to live," or in other words, to have an affair.
We have to admit, we find Dolly really likable here. Kitty doesn't want to meet Anna because she's uncomfortable with Anna's moral wrongdoing, but Dolly is able to sympathize. She recognizes that Anna is human, after all.
Of course, that's not exactly the lesson that Tolstoy wants us to carry away from this book. He gets that Anna is attractive and sympathetic, but she's still immoral in his opinion. So Dolly (and we, the readers) have to learn that Anna is a negative example of how to live a life. And the quickest way to prove to a natural mother like Dolly that Anna is no model for her to follow, is to show that Anna has lost her own connection to motherhood.
When Dolly visits Anna, she's impressed by all the grand furniture and high style that Anna lives in. But she also finds Anna's life to be hollow, and Dolly misses her own kids. All of her frustration with motherhood seems like heaven once she's away from the children she loves. And when Dolly finds out that Anna has chosen to have no more children so that she can stay attractive to Vronsky, well—that's when Dolly realizes that Anna's life is something she can't envy.
If there's one thing in the world Dolly loves, it's her kids, even if they frustrate her sometimes. The idea of willingly cutting yourself away from kids to have an affair is one of the worst things Dolly can imagine a woman doing. After this revelation, Dolly goes home to her children and feels relieved that she stayed with Stiva and never followed in Anna's footsteps.