Star Trek has never been a show to shy away from important issues. From Star Trek has always been there with a smart, intrinsically human answer. And why Star Trek is so loved and revered is because of how it tackles those issues.
The strength of the show and its characters comes from a shared moral code, their promise to do no harm, to help others in need, and to explore but not interfere. of Enterprise are bonded by a force that they all believe in, a force of good. This challenges them to make tough decisions and tackle deep issues involving themselves, the people they meet, and life in far-off reaches of the galaxy.
Fear
It must be scary being on a starship sometimes, hurtling through the sky into the great unknown. They have a ship counselor for a reason. While Enterprise is thrust into frightening situations often, some events will shake the crew to their core. When Dr. Crusher, who is scared of solitude, finds herself completely alone on Enterprise, it's a frightening jolt to overcome.
After an experiment gone wrong, Dr. Crusher notices the crew disappear one by one until she is completely alone. Scared at first, Crusher shows real strength and perseverance in a time of extreme loneliness and fear, in order to escape being swallowed up into the unknown.
Anxiety
Lt. Reginald Barclay is a nervous and anxious engineer that is on the brink of being transferred off the ship. His anxiousness causes him to stutter and perform underwhelmingly, even though he is live out his fantasies without his usual nervousness. The crew makes fun of him behind his back, but when the Captain realizes this problem and sees how talented Barclay can be, he and the crew make efforts to befriend Barclay and make him feel more comfortable on the Enterprise. This shows how comion without judgment can help someone break out of their shell and succeed.
Love And Relationships
Finding love in space must be a difficult task for any Starfleet member. Even on a ship full of people, it's still easy to feel lonely. And every relationship comes with its own challenges, as viewers have seen in numerous TNG episodes. From Geordi falling in love with a holodeck program to the turbulent but joyful relationship of Miles and Keiko, life and love aboard Enterprise can be difficult.
Even Captain Picard and Beverly once had a relationship, which is still a touchy subject for both of them years after the fact. It just goes to show that love and relationships are still hard to figure out in the 24th Century (except for maybe Will Riker).
Grief
The death of a loved one or crew member is never an easy thing to get over, as much of the Enterprise crew finds out throughout the series. It's easy to over the death of unnamed crew on away missions, but when it is a core member of the crew, and someone loved by many, that is something that will take more time to process. The death of Tasha Yar at the end of Season 1 hit Data the hardest, as they had a close, and even intimate relationship in the past. Data does not exactly have "feelings" as humans do, but viewers can see that he does show a kind of sadness and grief, especially after it is revealed in later seasons that he keeps a holo-image of her in his room.
Sentience
Lieutenant Data, one of the most beloved characters in all of Star Trek, is of course an android, who is not capable of human emotion. But as a core member of the Enterprise, and friend to many, it comes as a shock to everyone when Starfleet wants to take Data away for study and disassembly. Captain Picard must then prove that Data is in fact a sentient life form, capable of as much or even more than a human is, in order to keep him alive and aboard Enterprise. What follows is an imioned trial where Picard proves that Data is more human than people think, capable of love, vanity, and friendship, who is free to make his own decisions as a sentient being.
Humanity
When does a human stop being human? This is a question that is posed by of the crew after finding a Borg drone that has been severed from the collective hive mind. While Picard and the crew harbor hateful feelings towards the Borg, especially after what was done to them, they have a moral quandary that must be thought out before they take action. The borg drone is slowly becoming an individual again, but can still be dangerous. Crew have the idea of planting a virus within the drone, so when he goes back to the collective he infects the whole network. But as it becomes more apparent that this borg, who is now named Hugh, is becoming more and more human each day, they must let him decide for himself what he should do.
Family
Fighting and arguing with family is something that all humans can relate to, as well as the deep love and affection that can come from strong familial bonds. TNG explores family issues throughout the show, with characters like Crusher having to deal with her own son as a crew member, Worf who has his young son thrust upon him, and Picard who is anxious about returning home to his family.
Every character deals with their family issues differently, and each comes to with their relationships in separate ways. But this exploration of these issues shows that everyone has different relationships with family, and that's okay.
When To Lend A Hand
The starship Enterprise has always been ready to help lend a hand to those in need. They routinely answer distress calls and help those less fortunate than them. But when it comes to species and civilizations that are less developed, the Starfleet Prime Directive comes into play. For example, in the Season 1 episode "Symbiosis", t. Picard gets in the middle of a deal between two rival species, the Onarans, and the Brekkians. But after finding out that the Brekkians are exploiting the Onarans by getting them addicted to substances that only they have, Picard must decide whether or not to break the Prime Directive to help an exploited species or to not interfere and abandon them to a life of addiction.
Capitalism
Earth in the 24th Century is a socialist utopia, where they have managed to eradicate hunger and want. Everything is provided for the people of Earth, with help from replication technology, and warfare has ceased to exist on the planet. Throughout the show, there is a lot of commentary and critiques on Capitalism itself, especially when the lives of humans are juxtaposed with that of Ferengi, a society solely based on acquiring profit, an obvious jab at human society in the present. Star Trek has and always will show the bright future of humanity, but only if we are to fix the problems of present-day Earth, and the underlying causes of them, such as Capitalism itself.
Are Humans Worthy Of Existence At All?
This is a question posed by the very first episode of The Next Generation, when a god-like entity, "Q", comes to put humanity itself on trial. Q argues that humans are a savage, inferior species, that has killed, pillaged, and sucked the Earth of its natural resources. While these are mostly true statements, Picard argues that all of that is in the past, that humans have progressed past warfare and hunger, and have come out as a peaceful, developed species now centered on exploration and improvement. Amongst these arguments are certain critiques of the human condition as it is now, as well as the things that must be done in the future to ensure humanity's progress.