The theme of consent is often explored through Fusions - whether it's Stevonnie reaffirming their desire to stay Fused, the literal forced Fusion experiments done by Homeworld, or Garnet being furious at Pearl because she tricked her into Fusing under false pretenses, the writers want to make it very clear how important consent is.
Self-loathing, martyrdom and insecurity help no one, and even if you feel that you deserve scorn, you should love yourself. Steven gets furious at Pearl for projecting her insecurities on Connie in "Sworn to the Sword," and the Crystal Gems recall with annoyance how Pearl would sacrifice her physical form for Rose Quartz constantly. Then he gets hit with it when Connie becomes furious at him for sacrificing himself to Aquamarine and Topaz, rather than fuse with her to fight them. Meanwhile, he learns that Pink Diamond (the true identity of his mother Rose Quartz) hated herself intensely for being a Diamond and mentally trapped in a semi-permanent Womanchild state, and destroyed her old identity to save the Earth and to become a better person. This caused undue grief for all the Gems, including the Diamonds, that loved Pink for who she was but had a terrible way of expressing it.
Sometimes, even if you have the best of intentions, you can hurt people and make things worse when you lack empathy and don't think about the consequences of your actions. Steven and Connie in season five don't speak for several episodes because Steven didn't understand (at first) that Connie is hurt about his attempt at self-sacrifice, and Steven can't get through to Lapis when she freaks out about the Diamonds returning to Earth to fight it. Then he finds out that his mother was very much the same; Pink Diamond sacrificed her old identity and faked her death to save the Earth, as a last-ditch attempt to prevent it from being colonized. While she saved the Earth, she couldn't rescue the other Gems on either side from being Corrupted, and the Cluster ensued as a form of revenge.
Someone always loves you, and You Are Better Than You Think You Are: you have worth, no matter who you are or what you do. The Crystal Gems are still mourning Rose long after she gave up her form to have Steven, especially after we learn that she had intense self-loathing as Pink Diamond, but they take the time to tell Steven how much they love him, and how much he means to them. Steven stops Pearl from projecting her insecurities on Connie during swordfighting lessons in Season 2, and in turn seeks to help Pearl find out who she is now that Rose is gone. Uncle Andy is a bigoted, prejudiced jerk to everyone, but Steven manages to bring out his soft side by saying he wants to be part of Andy's life and spend time with him. No one really likes Lars, as he's an insensitive jerk, but his parents and Sadie mourn and miss him dearly when they learn that he died and Came Back Strong in space, and has a long way to reach home. Part of the reason why Pink Diamond faked her death is because she didn't think that the other Diamonds cared about her. Instead, they were completely consumed with grief over the presumed death of their fellow Diamond and attempted to avenge her by trying to destroy the Earth and hunting down Pink's "Rose Quartz" identity.
"Open Book" helps teach that it's trivial for friends to fight over things like fantasy books by having Connie not be mad at Steven despite him liking the ending of "Destiny's End" even though she hated it. It delivers the message more when she says that it's just a book, showing that you shouldn't care about stuff like TV shows or a book series more than your friends.
"Say Uncle" has the message that you shouldn't judge others just because they're different, you don't understand them, or you don't like their sense of humor or personality and that you shouldn't just blindly hate things.
"Love Letters" gives three lessons:
Love takes time and work. It's important to at least know who your partner is before rushing into a relationship with them.
You are in your own rights to reject someone if they are interested in you and you don't feel the same way.
Sometimes, Brutal Honesty is the best way to go, as letting someone down gently might give them the wrong message.
"We Need To Talk" deals with the problems of rushing into a relationship, and how a relationship can't work unless you both know and communicate with each other.
Stevenbomb 3 shows an uncommon lesson in cartoons: not all conflicts will be resolved by the end of the day. If someone is seriously upset about something that you did, it's going to take some time for them to forgive you. It also seems to preach several things about interpersonal conflicts:
"Cry for Help" is about how easy it is to mess up a relationship over something trivial to you, but serious to another.
"Keystone Motel" looks at the process of solving interpersonal conflicts; Ruby doesn't want to forgive Pearl yet, citing her rage at how much she feels betrayed by Pearl. Sapphire wants nothing but to forgive Pearl and move on, ending the conflict immediately. Both of these perspectives are revealed to be bad by the end of the episode; Ruby just wanted an excuse to be angry and not have to forgive anyone, while Sapphire neglected her own personal feelings for the sake of pragmatism. In the end, they reconcile, and come to the conclusion that while they will eventually forgive Pearl, they aren't ready to just yet. You have a right to be angry at someone for something they did, but in the end, keeping a conflict alive will only cause more turmoil, while ending it before everyone is ready to forgive will satisfy no-one. Furthermore, it also shows what happens when one tries to escape the conflict; Garnet accompanied Steven and Greg on their roadtrip so she could get away from the house. After she splits, both Ruby and Sapphire end up causing emotional turmoil for Steven and some financial troubles for Greg. Despite Greg's valiant efforts, Steven reaches his limit before he just lets out his emotional frustrations. It's a reminder on how handling strife affects everyone else.
"Historical Friction" has the message that everyone makes mistakes, and the important thing is to learn from said mistakes and keep trying. An especially important message considering the events of "Cry for Help".
The episode's play could also be considered an aesop on presenting historical figures in an honest light rather than glossing over their flaws. Considering how much certain historical figures are held on a pedestal without acknowledging the mistakes they made or the less-than-noble things they've done, this message couldn't be dropped too hard.
"Friend Ship": Pearl's constant attempts to get Garnet to forgive her only make the situation worse for everyone, leading to the Aesop of "people will forgive when they are ready, and trying to force someone to forgive another is wrong." There's also a secondary one: talking about your weaknesses enables healthy communication and understanding; Pearl's issues that caused most of the conflict are revealed to have come about because she saw Garnet as "the perfect relationship" and as a source of strength with no weaknesses that Pearl herself could never achieve because she's "just a Pearl". Garnet responds by saying that she does have weaknesses but she has to work in spite of them for the group's sake and that she draws strength from Pearl, giving several reasons why.
"Nightmare Hospital" and "Sadie's Song" both seem to have the message that you should talk to your parents about your problems and confront them if they're being controlling so you can work things out. They might not even realize how badly their behavior is affecting you until you tell them.
"Nightmare Hospital" also points out that while your parents might be strict and overbearing, refusing to show them what's going on in your life and what your interests are will only make things worse. While Dr. Maheswaran is forced to confront how much her Control Freak tendencies are hurting her daughter during this episode, Connie also has to realize that her increasing resentment and hiding the fact that she is fighting monsters and is secretly taking sword-fighting lessons is not doing anything to help. The two of them can only reconcile and massively improve their relationship when they actually talk things out and be honest with each other.
"Onion Friend": Even those who seem eccentric or weird can be great friends once you get to know them.
"Coach Steven": Devotion and awareness are key points to being strong, and strength is more than just physical.
"Story For Steven": After Greg's former manager Marty makes a rather objectifying comment suggesting Greg should ditch "one huge woman" (Rose) for "multiple smaller ones", Greg responds back proclaiming that women are people.
"Beach City Drift":
If you devote your time and energy towards hating someone, they'll waste your life without even doing anything.
If you're dealing with somebody who is a narcissistic jerk, the best thing you can do is to simply move on from their actions and be the better person for not letting them get to you. In the middle of the race, Stevonnie unfuses because Steven's hatred towards Kevin overwhelms them. Connie helps Steven realize that Kevin's getting to him by asking if beating him is actually worth it. Steven realizes that it's not, and Stevonnie refuses and spends the rest of race simply enjoying themselves, and is a Graceful Loser when Kevin beats them. Kevin is completely bewildered by this and can't wrap his head around it.
"Rocknaldo" makes a really important point: If you're a kind, selfless person, don't let other people use that as an excuse to walk on you or otherwise take advantage of your kindness. Anyone who does that is probably a Jerkass and deserves to be called out for it.
"Alone at Sea" has a pretty obvious metaphor for abusive relationships, and how breaking free of them is important, while emphasizing the guilt that abuse victims often feel (even though the relationship shown is strangely two-sided, with both individuals being poisonous to each other. The difference being that one of them actually feels bad about it and refuses to go back to it.).
Steven's confrontations with Bismuth, Jasper, and Eyeball during Season 3 force him to learn that while there's nothing wrong with trying to resolve things peacefully, diplomacy will not work with every foe you face, and sometimes Violence Is the Only Option, especially when dealing with somebody that will stop at nothing until they see you dead. However, by the same token, "Bismuth" is very adamant about making it clear that reckless, indiscriminate violence isn't the answer either, even if you're a Well-Intentioned Extremist.
"Mindful Education", especially the song "Here Comes a Thought", is about how refusing to think about something bad you did will only make things worse. Connie is only able to get past accidentally hurting a classmate when she deals with it and apologizes to him. Also, Steven's guilt over what happened with Bismuth, Jasper, and Eyeball almost overwhelms Stevonnie; he has to acknowledge that it happened and that he did the best he could in those situations to work through it.
It can be uncomfortable hanging around bigots and reactionaries (especially if they're your relatives) but they're still people, and they are capable of becoming more openminded when given the time and encouragement to do so.
Family, and life in general, always changes. It sucks, but you need to acknowledge that you can't stay in the past forever.
Your extended family may be weird, but that doesn't mean you can't love them.
"The Zoo"; You have to establish boundaries and consent that makes you comfortable in a relationship, regardless of what the other person might think or how they might react. Although all the Zoomans become miserable over Greg turning them down during the Choosening Ceremony, Greg still has every single right to do so.
Connie being furious at Steven for giving himself up to Homeworld and his Dissonant Serenity attitude after he made it back on a stroke of luck and not talking to him from for several weeks demonstrate that if you purposely and recklessly endanger yourself, even if it's done to protect somebody, people will be angry with you no matter how "justified" the circumstances made it because they're not going to be okay with being safe at the expense of you. Also, pretending that the emotional pain it causes (yours or theirs) doesn't matter is NOT okay because you're blowing off everyone's feelings, which doesn't help anyone.
Though it's more subtle, there's also the related point that sometimes, people can and/ or will help you with any burdens if you feel they're too big for you to take on by yourself (which constitutes Nanefua's whole campaign promise for handling future city issues as a community), and do care if you turn it down and end up hurting yourself worse (ie. Dewey taking responsibility despite his political incompetence, cynicism about city issues and election, and stubbornness to his and the city's detriment as mayor, despite running unopposed for his whole tenure).
"Back To The Kindergarten" teaches that even if something is damaged beyond repair and you're emotionally devastated from it (be it a home, a job, or a relationship), that doesn't mean you can't start over and rebuild from scratch, hard as that may sound. What's more, it teaches that the people who love you will be there to support you as you're going through a tough time.
"Your Mother and Mine" addresses that when looking at past events, especially controversial ones, there will be inevitable bias on all sides, whether from personal belief or from lack of knowledge. However, it's still important to gather them, as they do contribute to the event as a whole.
"What's Your Problem" dishes out three very good lessons:
Theres nothing wrong with looking out for other people, but at the same time you have to take care of yourself because your needs and issues are just as important, and you are not going to be able to help anyone if you can't help yourself once in a while.
You are not responsible for other people's problems and actions, and blaming yourself for them is extremely unfair and emotionally unhealthy to yourself.
Amethyst deciding that she won't dump "another thousand year old complex" onto Steven or anybody else demonstrates that always unloading your emotional baggage on someone will eventually weigh them down and there eventually comes a time where one has to mature up and solve their own problems more often.