In the commentaries for the Season one DVD, we learn that the J in Homer's full name was inserted as a tribute to Bullwinkle J. Moose.

This episode marks the first appearence of Otto, Jasper, Chief Wiggum, Smithers, and Wendell.

Chalkboard Gag: I will not skateboard in the halls.

Couch Gag: The family rushes in and squeezes onto the couch, causing it to collapse.

A three-eyed fish (which is named Blinky in a later episode) swims near the nuclear plant in a pond.

Sherri: Hey, Bart. Our dad says your dad is incompetent.
Bart: What does "incompetent" mean?
Sherri: It means he spends more time yakking and scarfing down donuts than during his job.
Bart: Oh, okay. I thought you were putting him down.

Homer: You'll get that punk someday, Moe.
Moe: Oh, I don't know. He's tough to catch. He keeps changing his name.

(Homer makes a proposal at City Hall.)
Homer: Ladies and gentlemen, esteemed councilmen, boys and girls, retired people with nothing better to do. Danger comes in many, many forms, from the dinosaurs that tormented our cavemen ancestors, to the--
Councilman: Simpson, get to the point!
Homer: I think we should put a stop sign at "D" street and 12th. The other--
Councilman: All in favor?
Councilmen: (In unison) Aye.
Councilman: Approved. Meeting adjourned. Coffee and maple logs in the lobby.
Homer: (In awe) Wow. They listened to me.

(Homer and a co-worker take a coffee and doughnut break at the power plant.)
Homer: (Chewing on a doughnut.) You know, I defy anyone to tell the difference between these doughnuts and the ones baked today. Hey, my boy's supposed to be here any second on a field trip. They been through here yet?
Co-worker: Come on, Simpson. If they wanted the kids to see you sitting around on your butt and stuffin' your face, they'd take them on a tour of your house.

(Bart prank calls Moe's.)
Moe: Moe's Tavern.
Bart: Is Mr. Freely there?
Moe: Who?
Bart: Freely. First initials "I. P."
Moe: Hold on, I'll check. Uh, is I.P. Freely here? Hey, everybody! I.P. Freely!
(Everyone at the bar laughs.)
Moe: Wait a minute. Listen to me, ya lousy bum. When I get a hold of you, you're dead. I swear I'm gonna slice your heart in half!
(Bart and Lisa laugh.)
Homer: You'll get that punk someday, Moe.
Moe: Ah, I don't know. He's tough to catch. He keeps changing his name.

During the montage of Homer making newspaper headlines with his signs, Homer is shown pointing at a diamond shaped "Sign Ahead" sign. But when the picture is taken and shown in the newspaper the sign becomes square shaped.

When Sherri and Terri tell Bart that their dad says his dad is incompetent, one of the twins has no body below her neck.

The fish swimming in the pond looks a lot like Blinky from Two Cars In Every Garage and Three Eyes On Every Fish.

In this episode, Smithers is dark-skinned with blue hair, due to a coloring mistake. In all future episodes, Smithers would be yellow-skinned with brownish-gray hair.

Odyssey

The episode title Homer's Odyssey is a reference to the Greek epic The Odyssey by Homer, where a character named Odysseus tries to return home from war. Read more about the book in the wikipedia article.

Get Smart

Right after Smithers shows Bart's class the video, he leads the class on a tour and they walk through doors which open very much like the doors that open during the opening sequence of the 1960's TV show, Get Smart.