When Shizuku is looking for a new book to read in the library, you can see on the bookshelf a book titled "Totoro", this is a reference to one of Hayao Miyazaki's earlier films My Neighbor Totoro (1988).
This marked the first use by Studio Ghibli of digital composition, meaning that elements of a scene were composed using a computer. In this case, it's the flying scene with Baron within Shizuku's story. The scene contains a lot of elements moving independently, including the small "planets" and Shizuku's characters. Although all these elements were animated by tradition means, they were combined using computer technology.
The backgrounds, clouds, and flying islands (they are called "Laputa") in the Flying sequence were painted by an artist named Naohisa Inoue. He is a former high school art teacher, who continued painting pictures of "Iblard" on his own. When he had his exhibition, he sent an invitation to Hayao Miyazaki, since Inoue was a huge fan of Miyazaki anime. Of course, he didn't expect anything, but Miyazaki showed up, and he bought one painting (which is now on the wall of the cafeteria at Ghibli). The painting is called "Upward Draft", and looks exactly like the fantasy scene in this film. Then, Miyazaki asked him, "Can you work on our next film?". Inoue also worked in this film as a voice actor. He did one of Mr. Nishi's friends, who joined the "Country Road" jam session (he's the tall one).
During the filming for the earlier Kiki's Delivery Service (1989), there was concern that the name of the film, along with the character "Jiji", a black cat, would create a trademark conflict with Kuroneko Takkyubin (Black Cat Express Delivery). Kuroneko Takkyubin is like UPS or FedEx in the US and has a black cat as its logo. A compromise was reached when Kuroneko Takkyubin agreed to act as a sponsor for the film. "Mimi wo Sumaseba/Whisper of the Heart" (1995), set in 1990s Japan, features "cameos" by several of Kuroneko Takkyubin's distinctive green and tan delivery trucks, including a shot of one in traffic early on, when Shizuku arrives at school, when she and Yuko are walking from school, and during the closing credits (where the logo is clearly visible). There may be many others in background shots.
"Take Me Home, Country Road" at the beginning of the movie was sung by Olivia Newton-John. The versions which Shizuku sang throughout the movie and at the end were sung by Youko Honna, the voice actress for Shizuku.
The trivia item below may give away important plot points.
During the closing credits, Shizuku and Seiji can be seen at the top of the screen, riding over the bridge on Seiji's bicycle. Later in the credits, Yuko meets Sugimura on the bridge and they walk off together. On the American (Disney) version of the DVD, these closing scenes are fully visible in the English-language version of the film. However, in the Japanese-language version on the same DVD, the top of the screen is cropped during the closing credits, cutting off the heads of the people passing over the bridge.