Throughout her travels Lucina meets and interacts with a lot of different people, forming different kinds of relationships with them. These relationships can vary from daughter, sister, wife, friend, and even mother. Because of this I am only going to focus on certain relationships which I feel are the most important to Lucina, otherwise this page could get really long.
Chrom, the prince of Ylisse.
First and foremost is Lucina's father, Chrom, the prince of the halidom of Ylisse and captain of the Shepherds militia. Despite being royalty he hardly seems like such and instead treats everyone with respect, regardless of stature. He cares strongly for his people and will put their needs before his own. In Lucina's time her father was killed––supposedly at the hands of someone he trusted. Although much of the details are hazy what is known is that he taught her how to use a sword and that he loved her dearly. When Lucina reveals herself to Chrom as his future daughter he is accepting of her right away, resulting in Lucina breaking down and crying in his arms.
Throughout the rest of the their time together Lucina is able to learn more about the kind of person her father was, resulting in her wanting to spend as much time as she could with him. However, because he already had a daughter in this time––his newborn baby Lucina––Lucina felt that she didn't belong and didn't want to be a burden to Chrom. Because of this Lucina promised that she would leave once peace had returned. However, Chrom thought no such thing and did not want to hear her say such words, assuring her she was not a burden. Instead he said he was grateful that she was born, that she grew into a fine and noble woman, and that she withstood terrible hardships and risked everything she knew to come to the past. Thanking her, Chrom told her that she was his daughter and friend, and that she'd always have a place at his side.
Robin, the mysterious tactician.
Another is the Robin, a tactician––who's name, appearance, and gender are all determined by the player––Chrom found unconscious on the side of the road. Having no memory of who they are, or where they came from it is soon discovered that Robin is exceptionally skilled at combat strategy. As such they quickly becomes an important and trusted companion to Chrom and the Shepherds. Although Lucina and Robin don't interact directly a whole lot––with the exception of Support Conversations––I still felt it important to include Robin here as there is more that connects them than you would think.
For me the highlight of their relationship is when Lucina witnesses Validar––the new king of Plegia and apparent father to Robin––controlling them to take the Fire Emblem from Chrom she realizes it was Robin who had killed her father in her time. To avoid losing her father again Lucina confronts Robin and decides the best way to prevent her future from happening is to kill them. Before Lucina can do this, however, Chrom will appear to stop her––but if Robin is her mother or husband Lucina cannot bring herself to kill them. When Chrom convinces her that their bonds are stronger now than they were in the future and that they will get them through the dark times together, Lucina begs Robin for forgiveness, and trusts that they prove her wrong and that the future would fall apart before their bonds do. Robin of course forgives Lucina.
Owain, the theatircal sword fighter.
Despite his crazy theatrics and what-have-you, Lucina's cousin, Owain, is another I feel worth mentioning. Reason being is Lucina actually UNDERSTANDS what he is talking about, and LIKES it, oftentimes preferring him to speak that way instead. With his story-type speech and arm that is always looking for a fight Owain is quite the character (to put it mildly), one that Lucina describes as "colorful". Owain also has a thing about giving his weapons and techniques names, claiming they bring out their potential, and making them an ally rather than just a simple tool. Whether that's true or not is uncertain, but it's an interesting thought nonetheless.
Aside from teaching Lucina the fine arts of his unusual speech, Owain also helps her with the upkeep of her Falchion, by repairing the hilt when needed as, unlike the blade, the hilt is susceptible to the ravages of time. The highlight of their relationship, however, is in their S Support where Owain gives Lucina a gift: a new sheath for Falchion. Although Lucina is confused by the gesture as her sword already had one he claimed that the current one was old and should be replaced. He then asks that it be his proxy for the times where he may not be able to aid her in battle. Happy, Lucina accepts and the two become partners with nothing to fear, Owain's abnormal talk and all. It's definitely awesome if nothing else.
Tiki, voice of the divine dragon.
Although she is an optional character and her role is relatively small I still felt Tiki worth mentioning here. Known by the people of Valm as The Voice, she is the daughter and oracle of the Divine Dragon, Naga. She is also a manakete––a dragon-like race––and despite her young appearance, she has lived since before the days of Marth, the Hero-King of old and as such, is very old (and has a tendency to sleep a lot as a result). When the two first met at her temple in Valm she mistakes Lucina for Marth due to their similar appearance, calling her "Mar-Mar"; a nickname she had given Marth during their time together, before being corrected by Lucina.
For me though, the highlight of their relationship is found within their Support Conversations. When Tiki comments on all the work Lucina does to help out her comrades she wonders if Lucina is trying to be like that of Marth. When Lucina admits that it is just something she does Tiki is surprised, but then tells her that it was a fateful decision of great importance when she took his name and wonders if the expectations that come with the name would crush her. Although Lucina is unsure of how to answer she is willing to do whatever it takes to ensure her friends' dreams are fulfilled. No easy task, I'm sure, but the fact that Lucina is willing to try is amazing and to hear those words from Tiki is a push in the right direction.