These are the relationships Penny has with characters in the show.
Boyfriend
Gumball
Gumball and Penny had both always been deeply in love with each other, but Penny always seemed to have had an easier time hiding her emotion (in other words, smoother) when Gumball was around. During Season 1 and 2, neither were capable of expressing it, even though they were both aware of the other's affection. So far, Penny has kissed Gumball on the cheek twice (the trailer for the series and "The End"); in addition, they attempted to kiss on the lips twice - once in "The Pressure," and another in "The Party." Both attempts failed as in the former, Banana Joe chopped down the treehouse that the kiss was taking place in, and in the resulting confusion, Gumball kisses Darwin on the lips in "The Pressure". In the latter, the moment was ruined by Richard honking his car horn loudly and interrupted the moment, so Gumball instead said to Penny that he loves her in "The Party," causing her to blush. Also in the episode "The Meddler," Penny and Gumball are shown to both love each other with Gumball not concealing this and the outcome of Penny thinking that all Gumball says and turns away with sad emotions. Later in the episode, Gumball fails the cheer leading tryouts and sacrifices his dignity for Penny, and in return, Penny forgiving him and concluding that Gumball likes her a bit. In "The Date," she had invited Gumball to her pet Mr. Cuddles' funeral, but Gumball mistakes it for a date and ends up embarrassing himself and losing Mr. and Mrs. Fitzgerald's respect for him. In the end, Gumball had found Mr. Cuddles in the bathroom, gaining back her, and her parents' respect. An example of her affection for Gumball that isn't important was in "The Mystery," when she cannot be spotted chasing after him angrily.
In Season 2, Gumball and Penny's relationship was explored further. In "The Knights," Gumball and Tobias both compete for Mr. Fitzgerald's permission to work on a medieval times project with Penny. Near the end of the episode, Tobias and Gumball "duel" for Penny's hand, and Tobias pushes her into the road. At the last minute, Gumball pushes Penny out of the way of her father's car, saving her life, and winning his respect once again. However, Mr. Fitzgerald crashes into The Wattersons' house, angering the family and thus reversing the situation for Mr. Fitzgerald. Again in "The Flower," Gumball competes for Penny out of jealousy for Leslie. His envy personifies itself into Gumball's soul, causing him to use very drastic measures to get rid of Leslie. Concerned for Leslie's well being (and Gumball's sanity), Penny, Darwin and Leslie resort to Carrie to exorcise Jealousy. After a lengthy cycle of Jealousy traveling for person to person, Gumball was freed of the entity. Soon afterward, Jealousy takes over Penny when Carrie gives Gumball a playful punch in the arm.
In Season 3, the relationship of Gumball and Penny escalated quickly starting off with the episode "The Kids" where Penny is shown to worry about Gumball and telling him why he didn't call her. Then the episode "The Shell" took this secret love to another level. In "The Shell," Penny was shown to break her shell open and revealing her true form. Gumball admired this new form but Penny took it the other way and thought that Gumball was in a traumatized by it. Through the episode, Gumball was clearly trying to protect Penny and at the end of the episode, they both confessed their feelings for each other, leading to the dating of them. In the next episode "The Burden," Gumball and Penny go on their first date and turns out to be a success and Penny telling Gumball he's probably marriage material. After that episode, "The Bros" is where Darwin was trying to interfere with their relationship. Penny took a break with their relationship just to help Gumball's relationship with Darwin. Penny and Gumball were back together as shown on "The Mirror" where Penny and Gumball are making flirting gestures across the classroom. This, however, ended when a misinterpreted message was passed down from Gumball to Penny. But in "The Spoiler," they are seen talking to each other on the phone.
In Season 4, in "The Romantic," Gumball sends Penny on a romantic quest in fear that he was losing her, which ends up failing and going erratic. After the quest, Penny forgives Gumball and tells him to tone down his romantic gestures and to not go so intense on their relationship, showing that Gumball and Penny have a crazy relationship but still together. In "The Misunderstandings," Penny went on date with Gumball. Near the episode end, Penny was choking but Gumball misinterpreted it and caused her to be mad at him, but he kissed her on the cheek.
Penny is very level-headed about their relationship as opposed to her boyfriend: whereas Gumball tends towards large and flashy ways of showing how he feels towards her, Penny is happy enough just being with the boy she loves.
In Season 5, the episode "The Petals" shows a brief moment of Gumball and Penny having a bit of a conversation, telling each other how much they love each other. While Gumball, Leslie and Darwin were performing the "frat-guy effect," Gumball was considered to be a medieval-pillaging creep. Penny said she would still love Gumball even though he looked like a medieval-pillaged creep, which the response was yes. Another instance of their relationship status in Season 5 was the episode "The Ex" where Penny was pretty much ignored by Gumball when he was focused on Rob, his arch nemesis. Penny and Gumball were not very communicative in this episode; for example, Gumball was blocking out Penny once he saw Rob. At the end of the episode, was sorry for Penny and said sorry in a very shocking way, with Alan having the words "I'm Sorry" on his face while stuck inside Penny's locker.
In Season 6, in "The Cringe," there are a collection of scenes involving flash forwards to Gumball and Penny's marriage, and eventually Penny's newborn baby in Gumball's arms. This signifies that most likely a successful relationship will be in the future for Gumball and Penny.
Family
Patrick
In "The Date," it is revealed that Penny's father is very protective of his daughter and at first does not approve of her hanging out with Gumball. In the Season 2 episode "The Knights," it is shown that he still does not care for Gumball, though he began accepting him at the end of the episode because of how he saved his daughter's life. But in "The Shell," Penny has appeared in many different forms, and presented as a "monster" to Patrick, which in return, made her run away from Elmore.
Mrs. Fitzgerald
Penny and her mother are likely to have a healthy relationship. In "The Date" and "The World," Penny is seen with her mom and the rest of the family. Her mom also watched her play, and even her attempt to kiss Gumball in "The Shell." Unlike her husband, she also seems to approve of Penny and Gumball being together as she allowed her to go to his house in "The Knights."
Polly
Penny has a sister named Polly in which the relationship is not that strong compared to Gumball. Polly was talked about in "The Romantic" and was shown in "The Grades." She is not yet out of her shell like Penny but has the similar appearance of Penny with her former peanut-shaped body. Penny has, at least, participated in Polly's grievance in Mr. Cuddles' funeral, and Polly has participated in Penny's birthday; those are the only indications of their family relationship.
Mr. Cuddles
Mr. Cuddles is the family’s pet spider, loved by all of the Fitzgeralds. In the episode "The Date," Mr. Cuddles' so-called funeral was present and Penny, Patrick, Pollie and Mrs. Fitzgerald attended this with Gumball. Mr. Cuddles is only shown two times in the series, "The Date" and "The World."
Leslie
In the episode "The Flower," the two begin to meet, making Gumball jealous. Later in the episode, it is revealed that the two are cousins. The two share a good relationship, as they are frequently seen hanging around together in the school cafeteria and the school playground.
Friends
Molly
The two appear to be friends. Penny was seen hanging out with her in her treehouse in the episode "The Pressure." It is unknown if Penny still remembers her as of "The Void."
Carrie
Penny and Carrie seem to be friends and were both seen in Molly's treehouse on at least a couple occasions.
Carmen
Penny and Carmen can be seen many times hanging out together. They are seen in Molly's treehouse, in the cafeteria, and in the bus. They are also together in the school's cheerleading squad. In "The Storm," Penny confronts Carmen about Gumball's supposed kiss with Carmen, and later helps her get Alan back from Masami.
Sarah
Penny and Sarah are seen to be talking to each other in the episode "The Cringe."
Sussie
Penny is a friend to Sussie, as shown in "The Bros" where Penny is giving Sussie chicken skins from her dinner inside a plastic container, knowing that Sussie has an enjoyable favor for meat and poultry.
Masami
Penny is also friends with Masami. Again, they hang out mostly in the cafeteria, or in cheerleading practice. In "The Storm" though, Masami zaps Penny when Penny attempts to ask her to "listen to her heart." Still, they get together with the other girls as seen in "The Bros."
Darwin
Penny and Darwin appear to have a friendly relationship. But overall, Darwin doesn't really talk to Penny much and generally interacts with her when she, him, and Gumball are hanging out together. However, he becomes jealous when she and Gumball start dating. After being spied on by him, Penny shows guilt rather than anger and sympathizes with him knowing that Darwin was doing all this because he was afraid of being alienated, implying that she was concerned for his well-being. She even helps Darwin and Gumball make up by making them vow by the bro-code, and afterwards, they themselves reconcile. After that, Darwin got over his jealousy and resumes his friendly relationship with her.
Unrequited admirers
Tobias
In the episode "The Knights," Tobias is shown to have a crush on Penny when Tobias is shown to be “fighting” for her. The feeling doesn't appear to be mutual because she only has feelings for Gumball. This is further shown in "The Move" when Tobias fails at flirting with Penny.
Tobias has tried (and failed) hitting on practically every girl in the school, turning his failure with Penny to be evident.