Since one of the authors of Good Omens was heavily involved in its TV series production, plenty of easter eggs, book-only jokes, and the general humor inherent to both Gaiman and Pratchett made it into the show. The upcoming second season, recently announced, will be based on an idea the two had for a sequel that never materialized, involving a lost angel in London.
Gaiman has said (via an interview in SyFy) that some background elements are homages to Pratchett and himself, as well as sci-fi properties they both liked, such as Doctor Who.
Aziraphale Pinned
Aziraphale has very different reactions to being pinned to the wall and threatened, first by demons and later by angels. Crowley comes after him in episode two for daring to claim that he's nice. However, Aziraphale doesn't look nearly as worried or startled as might be expected. He actually looks more contemplative. This scene certainly made everyone who sees Crowley and Aziraphale in a romantic relationship happy.
Later, when the angel Uriel pins Aziraphale to the wall, he actually does look scared. This is also bolstered by Aziraphale's nervous and shy behavior when any of the other angels show up, in comparison to the easy-going relationship he has with Crowley, shown through memorable quotes. Hanging out with Crowley for centuries - along with being the only angel permanently stationed on Earth - has changed Aziraphale's character so that he is less like the rest of the angels in Heaven. As Crowley says toward the end of the series, "We're on our own side."
Looking To The Left
When Crowley pops in on Aziraphale, he usually shows up on his left-hand side, which means Aziraphale is often looking in that direction before Crowley appears. When other angels like Gabriel pop in on Aziraphale, he also looks to the left, as if expecting Crowley. Aziraphale often starts those interactions happily, before realizing it's one of the angels and getting nervous again.
According to Gaiman's Tumblr, staging Crowley and Aziraphale like this throughout the centuries was intentional, though the why of it is up for debate. Gaiman mentions that the scene where the two swap back on the park bench after their trials seems subtlety wrong to a viewer, because they're used to seeing the duo sitting on opposite sides. The staging centers the characters in the audience's mind so that they even have proper places next to each other.
The Ritz (Or Is It?)
Crowley and Aziraphale dine at the Ritz in the sixth episode of the show, but astute viewers will notice that it isn't actually filmed there. Instead, it's the Criterion. This is noteworthy because director Douglas Mackinnon, who worked on the series Sherlock, was happy to add in some references (via SyFy). The Criterion is where Sherlock and John Watson first meet in the Sherlock Holmes stories by Arthur Conan Doyle. The creators of Sherlock also never actually got the chance to film at the Criterion, but the crew of Good Omens did.
Queen And The Bentley
One of the recurring running gags of the novel is that Crowley's Bentley will take any music tapes in it and turn them into a Best of Queen album. As an old car owned by a demon, it has a personality. After the final confrontation, the Bentley doesn't turn Handel's Water Music into anything else, and it's an element of the intensity of the climax and relief of averting the apocalypse that it stays as Handel's Water Music. This running gag is narration-heavy and fairly unadaptable without seeming clunky, so Crowley's Bentley just plays Queen all the time, including, wonderfully enough, when Crowley is barreling through the flaming M25. It's a background element that is really only noted in the book instead of the adaptation.
Video Games
Before Death meets the other Horsemen, he plays an arcade game. A quick shot of the high scorers list shows "T. Pratchett" in the number one slot. followed by Death in the next nine slots. This is a clear reference to Terry Pratchett, the other author of Good Omens, via Polygon.
It also translates to Pratchett beating Death, which is wonderful to think about, considering his own too-early death. Pratchett's Discworld novels are also famous for their innovative portrayal of Death, and people wrote him letters hoping the real death would be like the Discworld equivalent. It's easy to see similarities between Good Omens' Death and the Discworld version; they both speak in all capitals for one thing!
Sir Terry's Hat
Sir Terry Pratchett gets referenced quite a few times throughout Good Omens, and one easy visual cue is that his distinctive hat and scarf are hanging on a stand in Aziraphale's bookshop (via SyFy). They're easy to spot when Crowley-as-Aziraphale is looking around the newly-restored bookstore in episode six. Obviously, Pratchett couldn't be involved in the show's production, so putting his things in places he would like to go works well. The accessories aren't focused on, but they're in it for an eagle-eyed viewer to catch, which is part of the fun. Terry Pratchett is an undeniable part of Good Omens, and this proves it.
American Gods
Neil Gaiman's arguably most famous work is American Gods, which was adapted into a television series for Starz, with plenty of differences between the book and the show. When all the main characters roll up to the airbase to get to the climax of Good Omens, the soldier on guard duty is - blink-and-you'll-miss-it - reading American Gods. Not only does this imply that Gaiman exists in Good Omens' universe, but it also implies that he's just as successful. All told, this is a fun - and funny - background moment to find on a second watch. After all the references to Pratchett, it's nice for there to be a reference to Gaiman's own work as well.
Gallifrey
David Tennant is quite famous for playing the Tenth Doctor in Doctor Who. When Crowley is thinking of places to run away to, contemplating Alpha Centauri, one of the pages that float around him offers the suggestion of Gallifrey. Gallifrey, is, of course, the home planet of the Doctor and other Time Lords. Of course it makes sense for Crowley to go there, since he's technically already been there. This is a great Easter egg for fantasy fans who are also sci-fi fans. It is an intentional reference from Gaiman, since he and Pratchett enjoyed Doctor Who, along with Gaiman serving as a writer for a few episodes (via SyFy).
Elvis
In the book, Death mentions that he never laid a finger on Elvis Presley, implying that he isn't dead. Later, when introducing Raven Sable, the human name of Famine, there's a mention of someone who looks quite a lot like Elvis working in his kitchen. The show only keeps this latter element when we see a quick glimpse into the CHOW kitchen, but leaves out Death's dialogue regarding Elvis.
Therefore, this background detail really only works for book readers. since half of it is left out. Much of the minor content with the Horsepersons was left out for the sake of timing and budget. according to Express. Astute watchers might notice the Presley cameo, however.
Gaiman's Cameo
When Crowley is in the movie theater, there are a few other people sitting around him. One of them is a drunk, asleep, towards the front of the theater; this is Neil Gaiman's cameo. Originally, according to SyFy. Pratchett and Gaiman were going to cameo as other patrons at the sushi restaurant Aziraphale visits early on. However, after Pratchett's passing, Gaiman didn't feel right doing it alone, so asked for another scene where he could play a role. Director Douglas Mackinnon found one in the movie theater scene, and so that where Gaiman can be found.