I've seen it said a few times that Sam doesn't really have much of a character arc, and I want to talk about this because I thoroughly disagree -- on the contrary, I actually think he has multiple.
To have an arc, the character has to change in some way between the beginning of their narrative and the end. First of all, let's talk about some of the struggles that Sam faces throughout the game:
He has trouble standing up to people. We see this in his 4-heart and 6-heart events in particular. In the former, when he drops the egg and Jodi comes rushing in, he has a really hard time admitting to her what happened. He turns to you (the farmer) and asks you to tell her on his behalf. Similarly, in the 6-heart event, Mayor Lewis catches him skateboarding on private property, and asks for your opinion on the situation. If you join in to scold Sam, he apologises. If you stand up for him, he backs you up. In either event, he is not comfortable standing up for himself without your input.
He has trouble making decisions. We see this in all four of his first heart events. As outlined above, his 4- and 6-heart events involve him struggling to decide what to say for himself when he has done something perceived to be wrong/bad. In his 2-heart event, he can't decide what genre of music to make, and asks for your opinion. In his 3-heart event, he asks if he did the right thing by giving Vincent hope for their father's return. He may ask you on Wednesdays what subject he should write a song about. In all of these situations, he goes along with whatever you say, as long as you're nice and well-meaning about it. He also has a line of dialogue at >6 hearts debating whether or not to move out, but says "I hate making decisions".
He doesn't like disappointing people. This is similar to the above two points - he doesn't want to disappoint Vincent by giving him false hope, he doesn't want to disappoint the authority figures in his life by defying them or admitting that he's done the wrong thing - but there are several additional instances that demonstrate this. When he rejects you at the Flower Dance, he says "Hmm... I dunno. Maybe next year." as if he doesn't want to explicitly say no. If you invite him to see a movie with you that he doesn't like, he won't tell you outright that he doesn't like it; during the movie, he goes "oh! yeah... the movie's... interesting...", and afterwards he says "that one character was interesting... and um... heh". Like he's trying very hard not to disappoint you, because you went out of your way to invite him. He's not a pushover, but he's certainly a people-pleaser, and doesn't like to express negative sentiments outright.
He struggles to motivate himself to do things. This is relatively vague phrasing, but I think we see this manifest in a few quantifiable ways. Pre-marriage, Sam lives in a household where he's effectively still seen as a child. Jodi admits that the boys have never had to do chores. The result of this is that, after marriage, Sam has a lot of dialogue about not helping out around the farmhouse. He often says things like "Sorry I didn't make the bed", and "Maybe I'll help out on the farm some other day. I feel lazy today." He seems very aware of this and even possibly a little insecure, saying things like "You're not mad?" and "You know I'm sloppy... that's why you like me, right?". There are also lines like "There's too much stuff I want to get done today... Maybe I'll just watch TV instead." which ties his lack of motivation into his decision-making difficulties; there's too much to do, so he struggles to do any of it at all. It's also mentioned in Dev Update #12 that he often starts ambitious projects but doesn't finish them, of which the band is cited as one. In general, he seems to lack a sense of drive and purpose.
I think these traits can each be summarised with a single umbrella term: self-advocacy, decision-making, people-pleasing and motivation.
Now that we've defined our terms, let's take a look at Sam's later heart events, and how they address each of these core attributes.
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8-heart event: Sam and his band perform a gig in Zuzu City.
Decision-making: He has not only managed to settle on a genre and write songs, but has also taken tangible action by lining up a gig for his band. We don't necessarily know that he was responsible for organising the gig himself, but the fact that he got his shit together enough to put himself out there speaks to some level of character development.
Motivation: Mr "starts projects but doesn't finish them" has actually managed to take a project further than his bedroom. That is huge!!!
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10-heart event: Sam sneaks you into his room and lies to Jodi about you being there.
Self-advocacy: He has something he wants (his relationship with you), and instead of asking for permission or apologising to appease Jodi, he takes it for himself. He does this within the context of what is possible in his relationship with her, of course; given that he still lives at home and relies on Jodi for shelter + resources, it's not necessarily wise to admit something to her that he thinks will get him in trouble. The fact that she pretty much walks straight into his room and leaves the door open afterwards shows that he has a lack of privacy & autonomy in her household, and he is taking that back for himself in his own way by hiding you from Jodi. Being able to look her in the eye and lie to protect something he cares about is huge progress for someone who lives in a very tightly controlled household, and who has previously struggled to advocate for himself.
Decision-making: In a similar vein to the above, this is the first time we see Sam decide what to do in a situation without questioning it or seeking help from someone else. When Jodi leaves the room, he doesn't ask you "did I do the right thing by lying to her?". He has made that decision, and he sticks to it, because he knows exactly what he wants.
People-pleasing: Not to sound like a broken record here, but taking that step of lying to Jodi to protect something that's important to him shows huge progress in this area. Lying might disappoint her, and that would be rough, but it's a risk he's willing to take because he now has something (& someone!!) who makes that risk feel worth it. He also knows he has someone in his corner who will back him up, so it's a little less scary to stand on his own.
Motivation: He wants something (again, you), and takes the initiative to chase that. He reaches out to you with a letter, and he makes the first move in your relationship.
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14-heart event: Sam gets a job writing music for a children's TV show.
Decision-making: He decides to get this job himself. No one tells him to do it. He expresses to you that he wants to work, and assures you that even though he knows you make enough money to support the both of them, this is what he wants. You can give him advice, but he doesn't ask for it or seek your opinion at all. He keeps you in the loop because he wants you to be part of his endeavours, not because he needs you to decide anything on his behalf.
People-pleasing: Something worth noting is that, at the end of Sam's 8-heart event, he does not seem capable of taking full credit for the show. He shouts you out as the reason why his band was able to decide on a genre of music in the first place. He tells you several times afterward that he made a ton of mistakes, and was worried that you'd think he was a loser if he messed up onstage. The fact that his musical ambitions were only realised through the band is also, in itself, notable. In contrast, in his 14-heart event, he chases a goal that he wants, completely of his own volition. He shows you his song because he's proud of it, but he doesn't need you to tell him whether it's good or not; he knows he's made something great and he stands by that. He seems a lot more capable of deriving intrinsic motivation than he is earlier in the game, through means that he has independently decided are fulfilling.
Motivation: This is the big point addressed by this event. Notably, a lot of the points about Sam's lack of motivation are things that crop up after marriage. He marries the farmer, who is wealthy in their own right and does not really need him, so he ends up wandering aimlessly about the farmhouse and doing little things here and there to keep himself entertained. But his only real sense of purpose in this dynamic comes directly from his relationship with you, so when you're not around, it's difficult to motivate himself. Thus, he admits that he feels like he's gotten lazy, and wants to find work. This is him creating a sense of purpose for himself and (paraphrasing) "wanting to feel like he's contributing to something". The thing is, before marriage, Sam works at JojaMart, which he finds "depressing". There doesn't appear to be any reason why he couldn't have quit Joja and sought this job way earlier. He's been able to play guitar from the start, he's had a clear interest in music, and it's a job he can do remotely, so he wouldn't have even had to move out. On that basis, it seems like the only reason he didn't was because of that inability to intrinsically motivate himself. Once he's married to you, he has not only found a consistent source of extrinsic motivation, but has also become a lot better at drawing that motivation from within, and trusting his own instincts & choices. He is better at identifying what he wants and going for it. And that's HUGE.
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There are also a couple of other things that show development post-marriage, if you choose to pursue them. Firstly, having kids with Sam: he really steps up into that role, and in contrast to his earlier lines like "Maybe I'll help out on the farm some other day. I feel lazy today.", he has lines such as "I woke up early, fed the kids and changed their diapers! Weāre all set." That is a big departure from the unmotivated Sam we're used to, and shows significant character development thanks to a newfound sense of purpose.
Secondly, his summit cutscene dialogue: āI feel like I've changed a lot from when we first met... you know? ...you might call it 'maturity'... hehe. But I'm at a place, now, where I'm actually really excited about our future together.. Even if that means just doing normal stuff, like... uh... buying groceries and changing diapers... hah...ā. This one speaks for itself, but again, it's in line with that thread of Sam finding his purpose with you. Having someone who loves him and trusts him and cares about him does wonders for his vigour for life.
There's a general issue in Stardew wherein spouses' dialogue does not change after their 14-heart events, so even though Sam's arc is theoretically complete by this point, his dialogue still implies a certain listlessness while living in the farmhouse. I would chalk that up to awkwardness in the game's programming, since the dialogue and personalities are simply not that fleshed out after marriage - which makes sense, given that's not necessarily the main part of the game. However, going strictly by his heart events, Sam has so much development from your first meeting with him up to 14 hearts, where his self-identity and conviction therein is much more solidified.
