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. During season 1 and 2, neither were capable of expressing it, even though they were both aware of the other's affection, but Penny always seemed to have had an easier time hiding her emotions when Gumball was around. During that time, Penny had kissed Gumball on the cheek twice (in the trailer for the series and in "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, causing her to blush. Also in the episode "The Meddler," Penny and Gumball are shown to both love each other with Gumball accidentally saying that he does not love Penny which causes her to turn away crying. Later at the tryouts, although Penny doesn't perform well due to the earlier incident, Gumball fails the cheerleading tryouts and sacrifices his dignity for Penny, and in return, Penny forgives him and concludes that Gumball likes her after all.
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 Judith and Patrick's respect for him. In the end, Gumball found Mr. Cuddles in the bathroom, gaining back Penny and her parents' respect. In "The Party," Penny is jealous when she sees Gumball brought Tina for the dance. An example of her affection for Gumball that isn't important was in "The Mystery," when she and Darwin cannot be spotted chasing after him angrily like the rest of the students. Despite this, she has been seen in the out of character position, usually in oversight if she is just within a crowd who happen to be doing something, like in 'The Fight' egging on Gumball to fight Tina. But when she is focused on, it is done properly, like in 'The Sweaters', another case when she hears Gumball is going to be in a fight where her character is an actual part of it, showing him concern, holding his hand and hoping she can at least try to talk him out of fighting, just not knowing the full story. As he declares he has to fight, she goes along to support him, though it turns out to really be a tennis match.
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 affection, 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 Wattersons' house, angering the family, and thus reversing the situation for himself.
Again in "The Flower," Gumball competes for Penny out of jealousy of 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 asking him why he didn't call her. Then in the episode "The Shell," they finally became capable of expressing their feeling and became an item. In "The Shell," Penny was shown to break her shell and reveal her true form. Gumball was mesmerized by the beauty of her true form and literally rendered speechless, but Penny mistook Gumball's reaction for fear. Through the episode, Gumball was clearly trying to protect Penny and at the end of the episode, Gumball kissed Penny, therefore expressing their true feelings for each other.
In the next episode "The Burden," Gumball and Penny go on their first date and it turns out to be a success with 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 together again when they are 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 are still together. In "The Misunderstandings," Penny went on a date with Gumball. Near the episode's 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," Penny tells them that it makes them look like medieval-pillaging creeps. Gumball asks if Penny would still love him if he looked like a medieval-pillaging creep. Penny responds that she would and kisses him, and implies that Gumball is "the shining sun of her life."
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, Gumball said sorry to Penny in a very shocking way, with Alan having the words "I'm Sorry" written on his face while stuck inside Penny's locker and with her books vandalized with the words "I'm sorry Penny."
In season 6, in "The Cringe," there is 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. In "The Transformation" and "The Mess" she seems to trust him deeply, as in "The Transformation," she invites him to her house to settle an argument, and in "The Mess" she lets him babysit her younger sister, even though everything goes wrong. After Gumball and Darwin return Polly safely even though they are injured, Penny begins yelling and insulting Gumball.
In the series finale, "The Inquisition," Penny is one of the students who is turned human. Gumball accidentally kisses a girl who he thought was human Penny, but the real Penny angrily turns into her dragon form and shoots fire at Gumball, almost burning him alive. She cheers with the rest of the students when everyone is turned back to normal.
The series seems to leave the ultimate fate of their relationship ambiguous.
In the Gumball Chronicles episode, Vote Gumball and Penny, their relationship seems to be fixed as Gumball asks Penny to be his running mate for the elections for class president. Penny points out all of Gumball's previous flaws, but at the end says they are "totally good".
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 referred to as a "monster" by Patrick, which in return, made her run away from Elmore.
Judith
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 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. She is not yet out of her shell like Penny but resembles Penny with her former peanut-shaped body. Penny participated in Polly's grievance over Mr. Cuddles' death, and Polly participated in Penny's birthday. In "The Transformation," Polly is the only Fitzgerald who does not want Penny to return to her shell, although she does not seem to care that much either way.
In "The Mess," Penny lets Gumball babysit Polly, but he loses her, and when Polly is retrieved and the whole story comes out to Penny, she is furious, showing that she is quite a protective but caring older sister to Polly. She does not seem at all angry at Polly for wandering off, despite it resulting in Gumball and Darwin getting badly hurt. Indeed, Penny storms off with Polly and appears to be unsure if she should continue going out with Gumball, implying that she values Polly deeply, to the point that she considers breaking off her relationship with Gumball despite Polly not actually getting hurt.
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 held and Penny, Patrick, Polly, and Judith attended it 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 of occasions. They also work together to save Gumball in "The Flower."
Carmen
Penny and Carmen can be seen many times hanging out together. They are seen in Molly's treehouse, in the cafeteria, and on 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 several episodes.
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," however, 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 does not really talk to Penny much and generally interacts with her when she, him, and Gumball are hanging out together. However, in "The Bros," 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 is not mutual because she only has feelings for Gumball, and because she dislikes Tobias. 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, thus his failure to woo Penny is unsurprising.
Tobias, though, is seen talking to Sarah and Penny in the episode "The Pact," indicating that Penny might be neutral to Tobias.