What are the three parallel plot lines in Around the World In Eighty Days?

First of all, there is the main plot involving Phileas Fogg's circumnavigation of the globe in eighty days to win a bet. The two other subplots are, as one would expect, closely related to this one. Fogg makes his bet at the Reform Club the day after an audacious gentleman thief has robbed the Bank of England. It's the talk of the club, and Fogg's remark that it would be much easier these days for someone like the thief to hide himself somewhere in the world leads directly to the making of the bet.
This in turn leads to the second main plot line in the story: Inspector Fix's dogged pursuit of Fogg, believing him to be the very same gentleman thief who robbed the Bank of England. From his vast experience as a police officer, Fix has learned that thieves are invariably gentlemen. And as Phileas Fogg is a gentleman, especially one whose vast wealth is unexplained, Fix thinks he's put two and two together (wrongly).
Finally, we come to Fogg's romance with the Indian princess, Aouda. Fogg bravely saves Aouda from being chucked onto her late husband's funeral pyre in accordance with established custom. After this, a serious relationship starts to blossom between the two lovebirds. Indeed, towards the end of the story it becomes clear that it's his love for Aouda that means far more to Fogg than winning the bet. How do we know? Well, for one thing, Jules Verne helpfully spells it out in the title of the thirty-seventh and final chapter, during which Fogg gets married to Aouda:

In which it is shown that Phileas Fogg gained nothing by his tour around the world, unless it were happiness.