Young man in Paris, in love, and in a revolution
Because Sentimental Education is pretty long and covers roughly 20 years, there's some funny business with time. Years can pass in a few sentences. But Act I is really Frederick falling in love with Madame Arnoux and doing a lot of wooing, partying, and social climbing. While he's trying to get Madame Arnoux's attention, Frederick is trying to secure a mistress. (Hey, everybody's doing it). So he hooks up with Rosanette, who happens to be Monsieur Arnoux's mistress, too. (A character diagram may help here.)
She Friended Me
Things start heating up with Madame Arnoux. She's really shunning him, but finally—after she hears that he is engaged to someone else—she realizes that she loves him, too. They have a brief but passionate, um, friendship (strictly PG), until she misses a date with him. He goes off the rails for a while, getting engaged to a rich socialist (Madame Dambreuse) and getting his mistress (Rosanette) pregnant. But once all of those situations are dispatched, he goes on some travels.
Oh, Now You Show Up
Act III is short compared to the first two. It's more about the importance of what happens than about how long it takes to happen. Madame Arnoux, now a widow with white hair, shows up at Frederick's one day. They start to get romantic, but once he feels like she's the aggressor, he gets turned off. After 20 years of working it! So what does she do? "Gee, look at the time! I have to go. Goodbye." Fin.