Lemme start off by saying his entrance against Frieza was one of the, if not the greatest entrance of all time. It was extremely badass and a great way to introduce a new Super Saiyan and start things off for the saga. Now to get to the meaty stuff:
Trunks story starts off with the tragic death of his mentor and best friend Gohan. Because of this, Trunks wanted to rid the Android menace more than ever, to the point that it clouded his judgement when they were attacking a city near where he lived. Somewhat sharing his father's arrogance as well because of how powerful he believed he was. However, after the Androids had beaten him, like Bulma told him they would, he became more level headed and decided to go back in the past. This level headness and ending fights quickly and efficiently mentality persisted throughout his appearance afterwards. With the likes of Frieza, King Cold, Future Cell and Future Androids he always killed quickly and wasted no time toying with them or gloating. This is because his upbringing in his tragic world is where this mindset stemmed from.
One thing I notice people complain about Trunks is how whiny he was. Although I can understand why people disliked how Trunks complained a lot, I for one thinks it added to the strictly business mentality that he had. He was the one who experienced the horror of the Androids throughout his life so it's understandable that he'd be the most adamant to prevent anything similar happening. He complained about things that were completely justified, and it added to the character dynamics between Trunks and Vegeta. Vegeta did multiple heinous things throughout the saga which fuelled Trunks dislike for him, not bothering to save Baby Trunks was the final straw which led to Trunks even doubting his mothers claims that he had a good side to him which was hidden underneath his evil pride. Trunks still followed Vegeta, having hope (however small it may be) that Vegeta is the man his mother said he is, even going as far as to respect Vegeta's pride despite his dislike for him, which is a reflection of how much Bulma means to him. It showed how his mother is his true motivation, her words being what guided him throughout his story. If it wasn't for her, he wouldn't have believed in the things he did.
This is again shown when Trunks reveals his true reason for travelling to the past which was to create a world where the Androids didn't exist since his own Timeline can't be corrected and to have Goku find a weakness in the Androids. This further reflects how Bulma was his motivation because it was Bulma who wanted a better world regardless of if it's theirs, whereas Trunks was only interested in saving his world. It was Bulma who believed in Goku and worked on the time machine, when Trunks didn't. If it wasn't for Bulma, Trunks' wouldn't think too much of going to the past.
Another thing that adds to his character was his lack of hope in comparison to other characters, I find it to be as ironic as it is fitting since the whole purpose of his mission is to have hope yet he lacks it the most. i.e him not believing that a shutdown device would exist when Gohan did, and him not believing the fighters of the past would fair as well as they did. Again, this is justifiable since he grew up in a future filled with despair, and unlike Bulma who has great faith in Goku since she's experienced first hand all the things he accomplished, Trunks' didn't believe the Z-Fighters could make a difference against the Androids and only went to the past because it was his only option as well as Bulma's faith in Goku rubbing off on him and the fact that she was right about him being no match for the androids influencing him. Trunks must've learned not to cling onto hope too much after experiencing the horrors of his timeline, but his love for his mother spurred him into fulfilling her objective. After he fought Cell and lossed, he gave up knowing he had no chance of escaping or winning. I find it interesting how he was so ready to die because the life he lived could be a reason for that.
I found out that it was Trunks who wrote hope on the Time Machine instead of Bulma. This really adds to Trunks' character:
It symbolised what Trunks was going through at that point - The hopelessness he felt because of the Androids. Trunks wrote hope because he didn't feel the same way Bulma felt when looking at the time machine. He wrote hope in hopes of avoiding the sadness that comes from still feeling hopeless even when facing the pinnacle of hope which would've led to him giving up. To focus on the hope he still has regardless of how little it is and act on it instead of dwelling on his hopelessness. This is what his actions stemmed from (I.e Not telling Vegeta about his USSJ form). The time machine itself served as a reflection of Bulma and Trunks' relationship -- With Bulma's hope being great and the pioneer of change while Trunks' was little and only existed because of the pioneer.
When looked even deeper Trunks' hope adds to the son-father angle of his character. Much like Vegeta, he feared he'll live the rest of his life under those more powerful than him with the hopes of being freed from them decreasing as the years past and so developed arrogance to shield himself from the fact (Believing he'll beat the androids despite Gohan's failure) by being the descendent of Vegeta and the best hope as Gohan stated, by believing he's better than those before him. Trunks felt hopeless after losing his pride to the Androids post Gohan's death similar to how Vegeta felt no hope of regaining his pride after Goku died. They both had to cling on to something they had little of in order to keep going: Vegeta held on to his compassion and Trunks held on to his hope. However, Trunks still had the desire to fight and didn't let pride cloud his mind again because of his human nature, which Vegeta never had hence why Vegeta vowed to never fight again after his pride was shattered which adds to the themes of belonging that DBZ had with Gohan and Trunks being more fit to protect the earth than their fathers due to their human nature.
When Trunks went back to his time and told Bulma about how Vegeta fought for him, Bulma was in disbelief. This showed how curious Trunks was about his father and how Bulma couldn't bring herself to tell Trunks that he had no redeeming qualities, this was a surprisingly realistic mother-son angle. It further explains how Trunks insisted on following Vegeta despite his dislike for him.
Trunks was a superbly written character with great depth and development due to his relationship with Bulma and how that translated to his relationship with Vegeta. The history of Trunks storyline also added to the greatness of his character and his development.
