I know the hobbit movies make a big deal about how angsty teen Legolas is and completely screwed his relationship with his dad Thranduil. But whenever I read the Lord of the Rings I cannot help but think: my goodness, but Thranduil must have really spoiled Legolas rotten.
We first meet Legolas at the Council of Elrond. He is dressed simply and for light travel, and does not choose to wear his title as a Prince, instead identifying himself as a messenger from Thranduil, his king. Indeed he often chooses to call Thranduil his king rather than his father, which many have taken as a sign that relations between the two are strained.
However, I propose the direct opposite - that his many mentions of Thranduil as his Elven-Lord demonstrates not only his love for Thranduil, but also his respect and admiration of what a great king his father is for his people.
Consider this: whenever he mentions Thranduil or Mirkwood it is always with warmth and a subtle longing. He doesn't ever speak of his home in anything other than good terms, even if it has been marred and fallen into becoming Mirkwood (as opposed to Greenwood the Great).
And it isn't just to maintain pride in front of outsiders. When you contrast the way Boromir speaks of Gondor and his father the steward Denethor, to the way Legolas speaks of Mirkwood and Thranduil, it becomes clearer. Boromir is proud of his family, country and heritage and boasts of it. He claims Gondor is the main protector of Middle-earth, and that they have been fighting Mordor the hardest. But deep down he is questioning his father's rule, and is uncertain of Gondor's future.
Legolas is almost the opposite. He doesn't ever boast of his country, although Mirkwood, just like Gondor, is right at Mordor's doorstep. Instead he shares with the company personal anecdotes that show (not tell) exactly how good life is there despite the Shadow. He doesn't waver in his believe of his people's strength; indeed only he and Aragorn could face Galadriel without any fear or hesitation.
I think the reason why Legolas always mentions Thranduil as a king or lord is not because he doesn't want to acknowledge Thranduil is his father. It is that Legolas is so in awe of Thranduil and so respectful of him that whenever he is mentioned Legolas' first thought is not 'that's my dad and we're related', but instead is 'that's my king and my hero and he's so awesome'.
He doesn't consciously mention Thranduil as his father because his mind frame is not to introduce the company to Thranduil the way you would introduce your friends to your parents. He doesn't want them to think of him as just a nice uncle who is the father of a friend.
Legolas introduces Thranduil as his king because he wants the company to respect him as such, as the ruler of a great country who has done great deeds. He wants the company to see Thranduil as someone in-charge, as someone who is capable and should be given the proper courtesies accorded to him.
Some may question, in that case why doesn't Legolas call him 'my father the king'? It would highlight both Legolas' relationship to Thranduil as well as Thranduil's kingship.
I think this has to do with the circumstances as well as Legolas' humble personality. It is true that he doesn't ever make a big deal out of being prince, but at the same time this title means nothing much in the fellowship company. There is Aragorn, who would be king of Gondor. Boromir, eldest son of the ruling Steward of Gondor. Gimli, cousin of Balin who is the Lord of Moria. Gandalf, member of the Istari and councilor of many kingdoms. Peregrin, son of the Thain of Shire. Meriadoc, son of the Master of Buckland. Frodo, heir to Bilbo and a celebrity (of sorts) of the Shire.
Virtually everyone in the fellowship had a title (except Sam) and there is really no point in highlighting it among themselves when it doesn't matter. The only time where Legolas being Prince of Mirkwood would matter is when meeting other elves or when meeting allies of Mirkwood (like Dale for example). In which case in the former, Legolas being son of Thranduil is more than enough recognition for Elrond and Celeborn (as both of them were Sindar elves, and thus kin of Thranduil). And for the latter there wasn't any opportunity to do so.
Thus Legolas not mentioning the king is also his father is his way of downplaying his title as Prince, something which signifies his humility as well.
Finally, what makes me certain that Thranduil is a very caring father is Legolas' personality itself. Galadriel mentions that Legolas had always lived in joy, and this is very evident throughout the book.
In the entire journey, Legolas is shown to never despair. The only moment of fear was when he saw the Balrog (a famous elves bane) but otherwise he was also never fearful. This is despite the overwhelming circumstances the company finds themselves in. He always sees the bright side in every situation, frequently makes jokes and funny quips, and is in an overall joyful mood.
From the Hobbit, we learn that this cheerful outlook is shared by the Mirkwood elves, regardless of the growing Shadow. The only person who was affected at all was Thranduil himself, who was always wary of Mordor rising again. But he took careful pains to never affect his people, and they were always in good spirits despite the war that was happening.
I think Legolas was very well loved as a child, and always carefully sheltered. Not from the horrors and realities of war like death and destruction (because we can see he is a very capable warrior). No, he was instead sheltered from the despair and sorrow that would normally accompany death and destruction.
Legolas would see death and elves dying in battle, and Thranduil would tell him of the halls of Mandos and how death is just a path back to the Valar. He would look at the destruction of the woods and the land, and Thranduil would tell him it is like the passage of winter, but spring would come and the land would flourish again.
Legolas would have grown up knowing that even though bad things are happening, good things are still to come, and that's why he always has such hope in him and such joy. Unlike the mortal lives of Men and Dwarves, Elves are immortal and they can afford to wait for the end of the Shadow. There is no hurry, and because Mirkwood is untouched by any of the Rings, they don't need to worry about the power of the Three fading.
Legolas is happy and joyful because he is beloved, and I think that is a consequence of Thranduil's nurturing personality. It just makes much more sense when you compare him to the other characters who had lost both parents (Aragorn and Eomer), or had a stern and demanding father (Boromir and Faramir).
Legolas remains unchanged for the most part at the end of the journey because he was already in such a good place to start with. I think that is quite a comforting thought.