The ending of the first episode when Mike and the kids come to get Frankie after she was left stranded in the middle of nowhere by the woman she was trying to sell a car to who had three kids in jail.
Also, this line that Frankie says at the end:
Frankie: [voiceover] That's the thing about family. Sure, they eat your food and wreck your face and you've got to save them a thousand times a day from who knows what. And every now and then they save you.
The ending of "Thanksgiving" where it turns out that Frankie's family made thanksgiving dinner for her so it didn't have to be cancelled when she had to go to work for Mr. Ehlert that day so she didn't have to do all the cooking like she usually does.
In "Bad Choices", the Hecks aren't sure whether or not they should move into a Hickory Arms apartment. For example, Frankie seems to like Hickory Arms because it has more luxuries than their old home and their roof was broken, but Sue doesn't want to leave her friends and the other neighbors they knew. At the end of the episode, the neighbors come to help the Hecks fix the roof of their house, making Frankie realize that they truly belonged in this neighborhood that they were lucky to have to count on for help.
Frankie: [voiceover] You see, the thing we've forgotten is that we're part of the neighborhood. And the more our neighbors showed up to help us, the more Hickory Arms became a distant memory.
Axl's usually a total Jerkass, constantly complaining and barely ever showing affection. But in "The Break-up" he falls in love with a girl named Morgan. For days, he's bubbly because he's so in love. It's sweet to seem him so happy when he's usually so whiny and rude. Given the title of the episode, you'd know where this is going. Axl is heartbroken. Regardless of him being emotionally distant all the time, he breaks down and lets his mother hug him.
Frankie: Oh god. I forgot how soft his skin was.
In "A Tough Pill to Swallow", Sue forgets to renew her financial aid for college, and the only way to get into college is to pay the full amount. So Mike sells his half of the sports team diaper business to get the funds to put her through.