The main conflict in the story is Gurov's struggle to find meaning, truth, and beauty in a world that so often feels devoid of them. Gurov's brief dalliance with the titular lady with a pet dog has opened up new horizons for him, but so long as he remains stuck in his current life, he has no opportunity to explore them in any depth. To compound the problem, Gurov recognizes this as an ever-present conflict in most people's lives—one that they never seem to resolve.
By the close of the story, there's no sense that Gurov has managed to resolve this conundrum. He's got it into his head that the only way that he can find meaning, truth, and beauty in his life is by being with Anna, yet it's by no means apparent that this strategy for dealing with life's ugliness will succeed. The potential is there, undoubtedly, but as Chekhov makes clear right at the end, Gurov and Anna are only just beginning on the most difficult part of their relationship. That may be a resolution of sorts, but if so, it's a very strange one indeed.