These are the relationships Penny has with characters in the show.
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 seasons 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. 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 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 afterword, Jealousy takes over Penny when Carrie gives Gumball a playful punch in the arm.
In Season 3, in "The Shell," Gumball and Penny finally kissed on the lips for real, and so they are now dating. They go on their first date an episode later in "The Burden." Penny briefly leaves Gumball in "The Bros" because of guilt over supposedly stealing him away from Darwin. Although they seem to get back together by the end of the episode. Penny leaves Gumball again in "The Mirror" due to a curse that makes Gumball lose all his friends. Although, with the curse lifted, it can be assumed they are back together, they made a friendly phone call in "The Spoiler."
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. 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.
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.
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."
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.
The two appear to be friends. Penny was seen hanging out with her in her treehouse early in the series. It is unknown if Penny still remembers her as of "The Void."
Penny and Carrie seem to be friends, and were both seen in Molly's treehouse on at least a couple occasions.
Penny and Carmen can be seen many times hanging out together. First, they are seen in Molly's treehouse, then also in the cafeteria. 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.
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."
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.
In the episode "The Knights," Tobias is shown to have a crush on Penny. 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.