We first meet Michael Burnham and Captain Georgiou on a desert planet where they are attempting to save a new Star Trek race, the Crepusculans, from an 89 year drought. Georgiou affectionately refers to Burnham as "Number One," a term of respect given to many First Officers by their Captains. The first Number One in Star Trek was portrayed by the future wife of Star Trek's creator Gene Roddenberry, Majel Barrett, in original pilot for Star Trek, 'The Cage'. Barrett's Number One was also the first time a female First Officer was portrayed in Star Trek; Burnham is her successor in this regard. Commander Will Riker (Jonathan Frakes) was also often called "Number One" by Captain Picard on The Next Generation.
While on the Crepusculans' planet, Georgiou and Burnham attempt to free the water trapped under the planet while not making contact with the indigenous race. Burnham refers to Starfleet General Order One - the Prime Directive - which states that "No starship may interfere with the normal development of any alien life or society." Exposing a 'primitive' race to advanced alien technology may alter a planet and civilization's natural development - a big no-no in Star Trek. Of course, the Prime Directive has been violated multiple times in Trek, perhaps most infamously in the opening scenes of Star Trek Into Darkness when Kirk exposed the inhabitants of the planet Nibiru to the sight of the Starship Enterprise.
At the binary star system, Burnham volunteers to don an environmental suit to investigate the Klingon sarcophagus ship. Starfleet Officers leaving their ship to zip around in space in environmental suits occurred in both of the Star Trek films directed by J.J. Abrams and in Star Trek: First Contact when Picard, Worf, and Mr. Hawk (Neal McDonough) walked along the exterior of the Enterprise to battle the Borg. The moment Discovery's outer space scene with Burnham is most likely a homage to, however, is when Spock donned an environmental suit and flew into V-Ger in Star Trek: The Motion Picture.
When several starships arrive at the binary star system to back up the Shenzhou against the assembled Klingon fleet, one of the Starfleet ships is the U.S.S. Shran. Thy'lek Shran was an Andorian enemy, and later ally, of Captain Archer on Star Trek: Enterprise, portrayed by Jeffrey Combs. Combs also played the Vorta(s) known as Weyoun as well as the Ferengi Liquidator Brunt, arch rival of Quark (Armin Shimmerman), on Star Trek: Deep Space Nine. Extra bit of trivia: Combs was also a finalist for the part of Commander Will Riker on Star Trek: The Next Generation before Jonathan Frakes won the role.
Finally, though Starfleet's handheld phasers and communicators are pleasingly similar to those used in The Original Series, many hardcore Trekkers took umbrage to the standout piece of anachronistic technology on Discovery: the hologram projectors Captain Georgiou uses to communicate with Admiral Anderson (Terry Serpico). It is far too early for Starfleet to have holographic technology, according to established Trek canon. Ship-to-ship communication has traditionally been via viewscreens in Trek, and holodecks wouldn't be a staple on starships until the 24th century. The type of holographic tech seen in Discovery wouldn't be introduced until Deep Space Nine used it on board the U.S.S. Defiant during the latter stages of the Dominion War. This is one of many visual clues that Discovery may not be set in the Star Trek Prime timeline, as the producers claim, and may in fact take place in the Abrams movies' Kelvin timeline.
Did you spot more Easter eggs in Star Trek: Discovery's premiere episodes? Let us know in the comments.